All practice questions

Question 441Question →

A water purification plant processes raw water through three successive filtration stages: Stage A, Stage B, and Stage C. Stage A removes $\frac{3}{8}$ of the impurities present in the raw water. Stage B removes $40\%$ of the remaining impurities. Stage C removes $0.75$ of the impurities that remain after Stage B. If $4.5$ kilograms of impurities are successfully removed in Stage C, how many kilograms of impurities were originally in the raw water?

Show answer & explanation

Answer: 16

Answer

The original amount of impurities in the raw water was

16

kilograms.

The correct answer is

16

kg. To find this, we express the remaining impurities at each stage as a fraction of the initial amount

x

. After Stage A,

\frac{5}{8}x

remains. In Stage B,

40\%

is removed, meaning

60\%

of

\frac{5}{8}x

, or

\frac{3}{8}x

, remains. In Stage C,

0.75

(or

\frac{3}{4}

) of this remainder is removed, which is

\frac{3}{4} \times \frac{3}{8}x = \frac{9}{32}x

. Setting

\frac{9}{32}x = 4.5

gives

x = 16

.

Step-by-Step Solution

1

Represent the initial mass of impurities in the raw water with a variable.

Let

x

be the initial kilograms of impurities.

Establishing a variable allows setting up an equation to track the changes through each stage.

2

Calculate the remaining impurities after Stage A.

Impurities remaining =

x - \frac{3}{8}x = \frac{5}{8}x

kg.

Stage A removes

\frac{3}{8}

of the initial impurities, so

1 - \frac{3}{8} = \frac{5}{8}

of the impurities remain.

3

Calculate the impurities removed and remaining after Stage B.

Impurities removed in Stage B =

0.40 \times \frac{5}{8}x = \frac{1}{4}x

kg. Impurities remaining after Stage B = \frac{5}{8}x - \frac{1}{4}x = \frac{3}{8}x$ kg.

Stage B removes

40\%

of the impurities that remained after Stage A. Subtracting the removed portion from the starting amount for this stage yields the remaining fraction.

4

Express the impurities removed in Stage C.

Impurities removed in Stage C =

0.75 \times \frac{3}{8}x = \frac{9}{32}x

kg.

Stage C removes

0.75

(or

\frac{3}{4}

) of the impurities remaining after Stage B.

5

Equate the Stage C expression to the given value and solve for

x

.

\frac{9}{32}x = 4.5 \implies x = 4.5 \times \frac{32}{9} = 16

kg.

The problem states that

4.5

kg of impurities are removed in Stage C, so solving this equation yields the initial value.

Key Concept

Solving multi-step word problems involving successive applications of fractions, decimals, and percentages.

Alternative Method

Working backwards from the final stage can simplify the calculations. Since Stage C removes

0.75

of the impurities remaining after Stage B, and this amount equals

4.5

kg, the amount remaining after Stage B is

\frac{4.5}{0.75} = 6

kg. Since Stage B removes

40\%

of the impurities remaining after Stage A, the

6

kg represents

100\% - 40\% = 60\%

of the impurities remaining after Stage A. Thus, the amount remaining after Stage A is

\frac{6}{0.60} = 10

kg. Finally, since Stage A removes

\frac{3}{8}

of the original impurities, the

10

kg represents

1 - \frac{3}{8} = \frac{5}{8}

of the original impurities. The original amount is therefore

10 \times \frac{8}{5} = 16

kg.

Estimated Time:3m 0s

Question 442Question →

Two integers, $x$ and $y$ , are positioned on a number line. The distance between $x$ and $1$ is twice the distance between $y$ and $2$ . If the distance between $x$ and $y$ is exactly $5$ units, what is the sum of all possible values of $x + y$ ?

Show answer & explanation

Answer: 17

Answer

The sum of all possible values of

x + y

is

17

.

Representing the distances algebraically yields the system

|x - 1| = 2|y - 2|

and

|x - y| = 5

. Since

x

and

y

must be integers, splitting these equations into positive and negative cases yields exactly three valid integer coordinate pairs:

(13, 8)

,

(5, 0)

, and

(-7, -2)

. The sums (

x + y

) for these pairs are

21

,

5

, and

-9

, respectively. Adding these possible sums together gives a final total of

17

.

Step-by-Step Solution

1

Set up the absolute value expressions representing the distances.

|x - 1| = 2|y - 2|

and

|x - y| = 5

Distance on a number line between two points

a

and

b

is mathematically defined as

|a - b|

.

2

Split the distance condition

|x - y| = 5

into two coordinate cases.

x = y + 5

or

x = y - 5

An absolute value equation of the form

|A| = B

splits into

A = B

or

A = -B

.

3

Substitute

x = y + 5

into the first equation and solve for integer values of

y

.

y = 8

(which gives

x = 13

) and

y = 0

(which gives

x = 5

)

This generates the first set of valid integer coordinates satisfying all constraints.

4

Substitute

x = y - 5

into the first equation and solve for integer values of

y

.

y = -2

(which gives

x = -7

); the second algebraic option

y = 10/3

is discarded because it is not an integer

This generates the remaining valid integer coordinates satisfying all constraints.

5

Sum the value of

x + y

for all three valid coordinate pairs.

(13 + 8) + (5 + 0) + (-7 - 2) = 21 + 5 - 9 = 17

The question asks for the sum of all possible values of the expression

x + y

.

Key Concept

Using absolute value to represent distances on a number line and solving systems of absolute value equations under integer constraints.

Estimated Time:2m 30s

Question 443Question →

What is the sum of all integer values of $y$ that satisfy the inequality $|2y - 3| \le 7$ ?

Show answer & explanation

Answer: 12

Answer

The sum of all integer values of

y

that satisfy the inequality is 12.

Solving the inequality

|2y - 3| \le 7

requires setting up the compound inequality

-7 \le 2y - 3 \le 7

. Adding 3 to all parts gives

-4 \le 2y \le 10

, and dividing by 2 yields the interval

-2 \le y \le 5

. The integers in this closed interval are

-2, -1, 0, 1, 2, 3, 4

, and

5

. Summing these values gives 12, as the terms

-2

and

-1

cancel out with

2

and

1

.

Step-by-Step Solution

1

Set up the compound inequality

-7 \le 2y - 3 \le 7

An absolute value inequality of the form

|a| \le b

translates to

-b \le a \le b

.

2

Isolate the term containing

y

-4 \le 2y \le 10

Add 3 to all three parts of the compound inequality to eliminate the

-3

.

3

Solve for

y

-2 \le y \le 5

Divide all three parts of the inequality by 2.

4

Identify the integer solutions in the interval

-2, -1, 0, 1, 2, 3, 4, 5

The inequality includes the endpoints, so the integers satisfying the inequality are all integers from

-2

through

5

, inclusive.

5

Calculate the sum of the integers

12

Adding the integers:

(-2) + (-1) + 0 + 1 + 2 + 3 + 4 + 5 = 12

. The negative integers cancel out their corresponding positive counterparts (

-2

and

2

,

-1

and

1

).

Key Concept

Solving compound inequalities derived from absolute value inequalities and finding the sum of the integer solution set.

Estimated Time:1m 30s

Question 444Question →

A school band has between $100$ and $150$ members. When the band members line up in rows of $6$ , there are $2$ members left over. When they line up in rows of $9$ , there are also $2$ members left over. When they line up in rows of $5$ , there are no members left over. How many members are in the school band?

Show answer & explanation

Answer: 110

Answer

There are 110 members in the school band.

The number of members minus 2 must be a multiple of the least common multiple of 6 and 9, which is 18. The possible values between 100 and 150 are 110, 128, and 146. Among these, only 110 is divisible by 5, which satisfies the condition of having no members left over when grouped in rows of 5.

Step-by-Step Solution

1

Find the least common multiple (LCM) of the row sizes 6 and 9.

The LCM of 6 and 9 is 18.

Since the remainder is the same (2) for both row sizes, the number of members minus 2 must be a common multiple of 6 and 9.

2

Identify candidate numbers between 100 and 150 that are 2 more than a multiple of 18.

The candidate numbers are 110, 128, and 146.

To satisfy the condition of having a remainder of 2 when divided by 6 and 9, the total must be of the form

18k + 2

within the given range

[100, 150]

.

3

Check which candidate is divisible by 5.

110 is divisible by 5.

Since there are no members left over when lined up in rows of 5, the total number of members must be a multiple of 5.

Key Concept

Using the least common multiple (LCM) and remainders to solve divisibility word problems.

Estimated Time:1m 30s

Question 445Question →

In a school auditorium, $\frac{3}{8}$ of the total seats are in the balcony, and the remaining seats are on the main floor. For the spring play, $80\%$ of the balcony seats were occupied, and $60\%$ of the main floor seats were occupied. If there were exactly 117 empty seats in the auditorium, what was the total number of seats in the auditorium?

Show answer & explanation

Answer: 360

Answer

The total number of seats in the auditorium is 360.

The correct answer of 360 is found by expressing the empty seats from each section as a fraction of the total seats. Since the balcony represents

\frac{3}{8}

of the total seats and is

20\%

empty, the empty balcony seats equal

\frac{3}{40}

of the total seats. The main floor represents

\frac{5}{8}

of the total seats and is

40\%

empty, so the empty main floor seats equal

\frac{10}{40}

of the total seats. Summing these yields

\frac{13}{40}

of the total seats, which equals 117. Solving

\frac{13}{40}S = 117

gives a total of 360 seats.

Step-by-Step Solution

1

Define the variable for the total seats and find the fraction of seats in each section.

Let the total number of seats be

S

. The balcony has

\frac{3}{8}S

seats, and the main floor has

1 - \frac{3}{8} = \frac{5}{8}S

seats.

Establishing fractional partitions of the total capacity is necessary to set up the equation.

2

Calculate the fraction of total seats that are empty in the balcony.

Empty balcony seats =

20\%

of

\frac{3}{8}S = 0.20 \times \frac{3}{8}S = \frac{1}{5} \times \frac{3}{8}S = \frac{3}{40}S

.

Since 80% of the balcony seats are occupied, the remaining 20% are empty.

3

Calculate the fraction of total seats that are empty on the main floor.

Empty main floor seats =

40\%

of

\frac{5}{8}S = 0.40 \times \frac{5}{8}S = \frac{2}{5} \times \frac{5}{8}S = \frac{10}{40}S

.

Since 60% of the main floor seats are occupied, the remaining 40% are empty.

4

Sum the empty seat fractions and set the sum equal to the total number of empty seats.

Total empty seats =

\frac{3}{40}S + \frac{10}{40}S = \frac{13}{40}S

. The equation is

\frac{13}{40}S = 117

.

Combining the empty seats from both sections represents the total empty seat count.

5

Solve the equation for the total number of seats

S

.

S = 117 \times \frac{40}{13} = 9 \times 40 = 360

.

Multiplying by the reciprocal of the empty seats fraction isolates the variable for the total seats.

Key Concept

Solving multi-step word problems involving fractions, decimal conversions, and percentages.

Question 446Question →

On a standard number line, point $A$ is located at $-24$ and point $B$ is located at $6$ . Point $C$ is located between point $A$ and point $B$ such that the distance from $A$ to $C$ is $\frac{2}{3}$ of the distance from $C$ to $B$ . What is the coordinate of point $C$ ?

Show answer & explanation

Answer: -12

Answer

The coordinate of point

C

is

-12

.

Because point

C

with coordinate

c

lies between

-24

and

6

, the distance from

A

to

C

is

c - (-24) = c + 24

, and the distance from

C

to

B

is

6 - c

. Setting the distance from

A

to

C

to

\frac{2}{3}

of the distance from

C

to

B

gives the equation

c + 24 = \frac{2}{3}(6 - c)

. Multiplying both sides by

3

to clear the fraction results in

3c + 72 = 12 - 2c

. Collecting like terms yields

5c = -60

, which simplifies to

c = -12

.

Step-by-Step Solution

1

Express the distances between the points on the number line using their coordinates.

The distance from

A

to

C

is

c + 24

, and the distance from

C

to

B

is

6 - c

.

Since point

C

is positioned between points

A

and

B

, the inequality

-24 < c < 6

holds. This allows the absolute value distance expressions

|c - (-24)|

and

|6 - c|

to simplify directly to positive expressions without absolute value bars.

2

Formulate an equation based on the specified ratio of distances.

c + 24 = \frac{2}{3}(6 - c)

The problem states that the distance from

A

to

C

is

\frac{2}{3}

of the distance from

C

to

B

.

3

Solve the linear equation for the coordinate

c

.

c = -12

Multiplying both sides by

3

gives

3(c + 24) = 2(6 - c)

, which expands to

3c + 72 = 12 - 2c

. Rearranging the terms by adding

2c

to both sides and subtracting

72

from both sides results in

5c = -60

. Dividing by

5

gives

c = -12

.

Key Concept

Using absolute value properties to express distances on a number line and solving partitioning coordinate problems.

Estimated Time:1m 15s

Question 447Question →

A set of cards is numbered consecutively from $1$ to $N$ . If exactly $12$ of these cards have a number that is a multiple of both $6$ and $8$ , what is the greatest possible value of $N$ ?

Show answer & explanation

Answer: 311

Answer

The greatest possible value of

N

is

311

.

The correct value is 311. Since the numbers must be multiples of both 6 and 8, they must be multiples of their least common multiple, which is 24. The twelfth multiple of 24 is 288, and the thirteenth multiple is 312. To have exactly 12 such multiples, the maximum number N must be at least 288 but strictly less than 312, meaning the largest integer value is 311.

Step-by-Step Solution

1

Find the least common multiple (LCM) of 6 and 8.

The LCM of 6 and 8 is 24.

Any number that is a multiple of both 6 and 8 must be a multiple of their least common multiple.

2

Calculate the 12th and 13th multiples of 24.

The 12th multiple is

12 \times 24 = 288

and the 13th multiple is

13 \times 24 = 312

.

To have exactly 12 multiples in the set, the set must include the 12th multiple but exclude the 13th multiple.

3

Determine the maximum value of

N

such that 312 is not included.

N = 311

.

The largest integer less than 312 is 311. If

N

were 312 or greater, the set would contain 13 or more multiples.

Key Concept

Least Common Multiple and Divisibility Properties

Question 448Question →

On a standard number line, point $A$ has coordinate $-15$ and point $B$ has coordinate $17$ . Point $C$ is located to the right of point $B$ such that the distance between $A$ and $C$ is exactly $3$ times the distance between $B$ and $C$ . What is the coordinate of point $C$ ?

Show answer & explanation

Answer: 33

Answer

The coordinate of point

C

is 33.

The correct coordinate is found by setting up the distance equation for point

C

(with coordinate

c > 17

) relative to

A(-15)

and

B(17)

. The distance

AC

is

c - (-15) = c + 15

, and the distance

BC

is

c - 17

. Setting

c + 15 = 3(c - 17)

and solving yields

c = 33

, which is to the right of

B

.

Step-by-Step Solution

1

Define the variable for the coordinate of point C and write the expressions for distances.

Let

c

be the coordinate of point

C

. The distance between

A

and

C

is

|c - (-15)| = |c + 15|

, and the distance between

B

and

C

is

|c - 17|

. Since point

C

is to the right of point

B

(which is at

17

), we know

c > 17

, so

|c + 15| = c + 15

and

|c - 17| = c - 17

.

To set up an algebraic equation representing the physical distance relations on the number line.

2

Set up the equation using the given relationship.

The equation is

c + 15 = 3(c - 17)

.

The problem states the distance between

A

and

C

is

3

times the distance between

B

and

C

.

3

Solve the equation for

c

.

c + 15 = 3c - 51 \implies 15 + 51 = 3c - c \implies 66 = 2c \implies c = 33

.

To find the coordinate of point

C

.

Key Concept

Calculating distances between points on a number line using absolute value and solving the resulting equations.

Alternative Method

Use geometric visualization: The distance from

A(-15)

to

B(17)

is

17 - (-15) = 32

units. Since point

C

lies to the right of

B

, the distance

AC

is the sum of

AB

and

BC

. Therefore,

AC = 32 + BC

. We are given that

AC = 3 \times BC

. Substituting this gives

32 + BC = 3 \times BC \implies 2 \times BC = 32 \implies BC = 16

. Since

C

is 16 units to the right of

B(17)

, its coordinate is

17 + 16 = 33

.

Estimated Time:1m 30s

Question 449Question →

If $x$ is a real number such that $\sqrt{\frac{4^{x+2} + 4^{x+2} + 4^{x+2} + 4^{x+2}}{8^{x-3}}} = 64$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The value of

x

is

3

.

By rewriting the repeated addition in the numerator as

4 \times 4^{x+2} = 4^{x+3}

and converting all bases to 2, the expression inside the square root simplifies to

2^{15-x}

. Taking the square root gives

2^{\frac{15-x}{2}}

. Setting this equal to

64

(which is

2^6

) and equating the exponents yields

x = 3

.

Step-by-Step Solution

1

Rewrite the sum in the numerator

4^{x+2} + 4^{x+2} + 4^{x+2} + 4^{x+2}

as a product.

4 \times 4^{x+2} = 4^{x+3}

Adding a term to itself four times is equivalent to multiplying that term by 4.

2

Convert the base 4 numerator and base 8 denominator to base 2.

Numerator:

(2^2)^{x+3} = 2^{2x+6}

; Denominator:

(2^3)^{x-3} = 2^{3x-9}

Expressing terms with the same base allows the use of exponent laws to simplify the fraction.

3

Simplify the fraction by subtracting the denominator's exponent from the numerator's exponent.

\frac{2^{2x+6}}{2^{3x-9}} = 2^{(2x+6)-(3x-9)} = 2^{15-x}

The quotient rule for exponents states that

\frac{a^m}{a^n} = a^{m-n}

.

4

Apply the square root to the simplified fraction, set it equal to

64

, and express both sides as powers of 2.

\sqrt{2^{15-x}} = 2^{\frac{15-x}{2}} = 64 = 2^6

The square root of a term is equivalent to raising that term to the power of

\frac{1}{2}

.

5

Equate the exponents and solve for

x

.

\frac{15-x}{2} = 6 \implies 15-x = 12 \implies x = 3

If two exponential expressions with the same positive base are equal, their exponents must be equal.

Key Concept

Simplifying expressions using the laws of exponents and properties of roots

Question 450Question →

A rectangular field has a length of $1.5 \times 10^4$ meters and a width of $4.0 \times 10^3$ meters. If a tractor can mow $6.0 \times 10^5$ square meters per hour, how many hours will it take the tractor to mow the entire field?

Show answer & explanation

Answer: 100

Answer

It will take 100 hours for the tractor to mow the entire field.

The total area of the field is the product of its length and width:

(1.5 \times 10^4) \times (4.0 \times 10^3) = 6.0 \times 10^7

square meters. Dividing this area by the tractor's mowing rate of

6.0 \times 10^5

square meters per hour gives

\frac{6.0 \times 10^7}{6.0 \times 10^5} = 1.0 \times 10^2 = 100

hours.

Step-by-Step Solution

1

Calculate the area of the rectangular field.

Area =

6.0 \times 10^7

square meters

The area of a rectangle is found by multiplying its length by its width:

(1.5 \times 10^4 \text{ m}) \times (4.0 \times 10^3 \text{ m}) = 6.0 \times 10^7 \text{ m}^2

.

2

Calculate the time required to mow the field.

Time =

100

hours

Divide the total area by the tractor's mowing rate:

\frac{6.0 \times 10^7 \text{ m}^2}{6.0 \times 10^5 \text{ m}^2/\text{hour}} = 1.0 \times 10^2 = 100

hours.

Key Concept

Multiplication and division of numbers in scientific notation

Question 451Question →

A smart thermostat is programmed to reduce a home's heating energy usage. In January, the heating energy usage is reduced by $15\%$ compared to the baseline usage. In February, the usage is reduced by an additional $\frac{1}{8}$ of January's usage level. In March, the usage is reduced by $20\%$ of February's usage level. If the baseline heating energy usage was $400$ kilowatt-hours (kWh), what is the total reduction in energy usage from the baseline to the end of March, in kWh?

Show answer & explanation

Answer: 162

Answer

The total reduction in energy usage from the baseline to the end of March is 162 kWh.

The baseline usage of

400

kWh is reduced by

15\%

in January, leaving

340

kWh. In February, a reduction of

\frac{1}{8}

of

340

kWh reduces usage by

42.5

kWh, leaving

297.5

kWh. In March, a reduction of

20\%

of

297.5

kWh reduces usage by

59.5

kWh, leaving a final usage of

238

kWh. The total reduction is the difference between the baseline and final usage levels, which is

162

kWh.

Step-by-Step Solution

1

Calculate January's energy reduction and January's usage level.

January reduction is

60

kWh; January usage level is

340

kWh.

January's reduction is

15\%

of the

400

kWh baseline (

0.15 \times 400 = 60

kWh). Subtracting this reduction from the baseline gives January's usage level (

400 - 60 = 340

kWh).

2

Calculate February's energy reduction and February's usage level.

February reduction is

42.5

kWh; February usage level is

297.5

kWh.

February's reduction is

\frac{1}{8}

of January's usage level (

340

kWh), which is

0.125 \times 340 = 42.5

kWh. Subtracting this from January's level gives February's usage level (

340 - 42.5 = 297.5

kWh).

3

Calculate March's energy reduction and March's usage level.

March reduction is

59.5

kWh; March usage level is

238

kWh.

March's reduction is

20\%

of February's usage level (

297.5

kWh), which is

0.20 \times 297.5 = 59.5

kWh. Subtracting this from February's level gives March's usage level (

297.5 - 59.5 = 238

kWh).

4

Calculate the total energy reduction from the baseline to the end of March.

The total reduction is

162

kWh.

The total reduction can be found by adding the reductions from each of the three months (

60 + 42.5 + 59.5 = 162

kWh) or by subtracting the final usage level from the baseline (

400 - 238 = 162

kWh).

Key Concept

Applying sequential percentage and fraction reductions to changing base values.

Question 452Question →

On a vertical number line representing elevation in meters, a research drone is at coordinate $d$ and a submarine is at coordinate $s$ . The coordinate of the drone is a positive integer, and the coordinate of the submarine is a negative integer. The distance between the drone and the submarine is $150$ meters. If the absolute value of the submarine's coordinate is $4$ times the coordinate of the drone, what is the coordinate of the drone?

Show answer & explanation

Answer: 30

Answer

The coordinate of the drone is 30.

The correct coordinate of 30 is found by defining the distance between the positive drone coordinate

d

and negative submarine coordinate

s

as

d - s = 150

. Since

s < 0

, its absolute value

|s|

is equal to

-s

. Substituting

-s = 4d

gives

d - (-4d) = 150

, which simplifies to

5d = 150

and yields

d = 30

.

Step-by-Step Solution

1

Set up equations based on the problem description.

d > 0

,

s < 0

, distance

|d - s| = 150

, and

|s| = 4d

.

To translate the verbal description of elevations and distances into mathematical expressions.

2

Simplify the distance and absolute value expressions using the signs of the coordinates.

Since

d

is positive and

s

is negative,

d - s = 150

. Since

s

is negative,

|s| = -s

, so

-s = 4d

or

s = -4d

.

To eliminate absolute values based on the known signs of the variables.

3

Substitute the expression for

s

into the distance equation and solve for

d

.

d - (-4d) = 150 \Rightarrow 5d = 150 \Rightarrow d = 30

.

To solve the system of linear equations to find the drone's coordinate.

Key Concept

Using absolute value to represent distance on a number line and solving equations involving sign constraints.

Question 453Question →

The diameter of a human red blood cell is approximately $7.0 \times 10^{-6}$ meters, and the diameter of a typical influenza virus is approximately $1.4 \times 10^{-7}$ meters. How many times larger is the diameter of the red blood cell than the diameter of the influenza virus?

Show answer & explanation

Answer: 50

Answer

50

To determine how many times larger the diameter of the red blood cell is than the diameter of the influenza virus, divide the larger diameter by the smaller diameter:

\frac{7.0 \times 10^{-6}}{1.4 \times 10^{-7}}

. Dividing the coefficients gives

\frac{7.0}{1.4} = 5

. Dividing the powers of ten using the quotient rule for exponents gives

\frac{10^{-6}}{10^{-7}} = 10^{-6 - (-7)} = 10^1

. Multiplying these results gives

5 \times 10^1 = 50

.

Step-by-Step Solution

1

Set up the ratio of the larger diameter to the smaller diameter

\frac{7.0 \times 10^{-6}}{1.4 \times 10^{-7}}

To find how many times larger one quantity is than another, divide the larger quantity by the smaller quantity.

2

Divide the decimal coefficients

5.0

Dividing

7.0

by

1.4

simplifies the numerical coefficient.

3

Divide the exponential terms using exponent properties

10^1

Using the quotient rule for exponents,

\frac{10^a}{10^b} = 10^{a - b}

, so

10^{-6 - (-7)} = 10^1

.

4

Combine and simplify the final value

50

Multiplying the coefficient by the simplified power of ten yields

5.0 \times 10 = 50

.

Key Concept

Division of numbers in scientific notation using properties of exponents

Estimated Time:1m 30s

Question 454Question →

A scientist monitors the temperature of two research chambers. Chamber A is kept at $-5^\circ\text{C}$ and Chamber B is kept at $7^\circ\text{C}$ . The scientist sets a third chamber, Chamber C, to a temperature of $T^\circ\text{C}$ such that the distance between $T$ and the temperature of Chamber A on the Celsius scale is exactly $3$ times the distance between $T$ and the temperature of Chamber B. If the temperature of Chamber C is warmer than Chamber A but colder than Chamber B, what is the value of $T$ ?

Show answer & explanation

Answer: 4

Answer

The correct temperature value of Chamber C is 4.

The temperature of Chamber C must be

4^\circ\text{C}

because the distance from

4

to

-5

is

|4 - (-5)| = 9

and the distance from

4

to

7

is

|4 - 7| = 3

. The distance of

9

is exactly

3

times the distance of

3

. Furthermore,

4

lies between

-5

and

7

, satisfying the condition that Chamber C is warmer than Chamber A but colder than Chamber B.

Step-by-Step Solution

1

Represent the distances on the number line using absolute value expressions.

The distance to Chamber A is

|T - (-5)| = |T + 5|

and the distance to Chamber B is

|T - 7|

.

Distance between two points

x

and

y

on a number line is represented by

|x - y|

.

2

Set up the algebraic equation reflecting the relationship between the distances.

|T + 5| = 3|T - 7|

The problem states the distance to Chamber A is exactly 3 times the distance to Chamber B.

3

Solve the absolute value equation by considering both positive and negative cases.

Case 1:

T + 5 = 3(T - 7) \Rightarrow T = 13

. Case 2:

T + 5 = -3(T - 7) \Rightarrow T = 4

.

The equation

|x| = |y|

implies

x = y

or

x = -y

.

4

Verify which solution satisfies the temperature boundary condition.

Since Chamber C must be warmer than

-5^\circ\text{C}

but colder than

7^\circ\text{C}

, the only valid value is

T = 4

.

The value

T = 13

is warmer than both chambers and does not lie between them.

Key Concept

Using absolute value to represent distance on a number line and solving absolute value equations with boundary conditions.

Alternative Method

Alternatively, visualize this on a number line. The total distance between Chamber A (

-5

) and Chamber B (

7

) is

12

units. Since Chamber C lies between them and the distance from C to A is

3

times the distance from C to B, we can divide the

12

-unit interval into

3 + 1 = 4

equal parts. Each part is

12 \div 4 = 3

units. Chamber C is located

1

part away from Chamber B (towards Chamber A), which places it at

7 - 3 = 4

.

Estimated Time:1m 30s

Question 455Question →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $x^2 + y^2 - 8x - 6y = 0$ . A point on this circle has an $x$ -coordinate of $1$ and a positive $y$ -coordinate. What is the $y$ -coordinate of this point?

Show answer & explanation

Answer: 7

Answer

The positive y-coordinate of the point on the circle is 7.

Substituting

x = 1

into the circle's equation

x^2 + y^2 - 8x - 6y = 0

gives

1 + y^2 - 8 - 6y = 0

, which simplifies to the quadratic equation

y^2 - 6y - 7 = 0

. Factoring this equation yields

(y - 7)(y + 1) = 0

. The solutions are

y = 7

and

y = -1

. Because the y-coordinate must be positive, the correct value is 7.

Step-by-Step Solution

1

Substitute the x-coordinate

x = 1

into the circle's equation.

1^2 + y^2 - 8(1) - 6y = 0

This sets up the equation to solve for the corresponding y-coordinates.

2

Simplify the equation.

y^2 - 6y - 7 = 0

Combining the constant terms creates a standard quadratic equation in terms of y.

3

Solve the quadratic equation by factoring.

(y - 7)(y + 1) = 0

, yielding

y = 7

or

y = -1

Factoring allows us to find the two possible y-values that satisfy the equation.

4

Choose the positive y-coordinate.

y = 7

The question specifies that the y-coordinate must be positive.

Key Concept

Substituting a known coordinate into a circle's equation to find the other coordinate using quadratic equations.

Question 456Question →

A highway is constructed up a mountain pass at a constant incline. At a distance of $3.6$ miles from the starting toll gate, the elevation of the highway is $1,420$ feet above sea level. At a distance of $8.4$ miles from the toll gate, the elevation is $3,100$ feet above sea level. What is the slope of the highway's elevation profile, in feet per mile?

Show answer & explanation

Answer: 350

Answer

The slope of the highway's elevation profile is 350 feet per mile.

The slope represents the constant rate of change of elevation per mile. By defining the coordinates as

(x_1, y_1) = (3.6, 1420)

and

(x_2, y_2) = (8.4, 3100)

, we calculate the slope using the slope formula. Substituting the values gives

m = \frac{3100 - 1420}{8.4 - 3.6} = \frac{1680}{4.8} = 350

.

Step-by-Step Solution

1

Represent the given data as coordinate points where the x-coordinate represents the distance in miles and the y-coordinate represents the elevation in feet.

The coordinate points are

(3.6, 1420)

and

(8.4, 3100)

.

Slope is the change in the dependent variable (elevation) per unit change in the independent variable (distance).

2

Apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

to find the rate of change.

m = \frac{3100 - 1420}{8.4 - 3.6} = \frac{1680}{4.8}

The slope represents the constant rate of change of elevation relative to the distance traveled.

3

Perform the division to find the numerical slope.

350

Dividing 1680 by 4.8 yields the exact value of 350.

Key Concept

The slope of a line represents a constant rate of change between two variables, calculated as the change in the vertical coordinate divided by the change in the horizontal coordinate.

Question 457Question →

A manufacturer of solar panels inspects their production in three stages. In the first stage, $\frac{1}{12}$ of the total panels produced are identified as defective and recycled. In the second stage, $20\%$ of the remaining non-defective panels are found to have cosmetic flaws and are sold at a discount. In the third stage, $\frac{1}{5}$ of the panels that passed the first two stages without any defects or flaws are selected for quality testing. If $1,760$ panels passed the first two stages but were NOT selected for quality testing, what was the total number of panels originally produced?

Show answer & explanation

Answer: 3000

Answer

The total number of panels originally produced was

3,000

.

The correct answer of

3,000

is determined by calculating the cumulative fraction of panels that are not defective, do not have cosmetic flaws, and are not selected for testing. In Stage 1, subtracting

\frac{1}{12}

leaves

\frac{11}{12}

of the original panels. In Stage 2, subtracting

20\%

of those leaves

80\%

(or

\frac{4}{5}

), resulting in

\frac{11}{15}

of the original panels. In Stage 3, subtracting

\frac{1}{5}

of these leaves

\frac{4}{5}

, resulting in

\frac{44}{75}

of the original panels. Setting

\frac{44}{75}

of the original panels equal to

1,760

and solving for the total original panels yields

3,000

.

Step-by-Step Solution

1

Calculate the fraction of panels remaining after the first stage of inspection.

\frac{11}{12}

of the original panels

Since

\frac{1}{12}

of the total panels are defective and recycled, the remaining portion is

1 - \frac{1}{12} = \frac{11}{12}

.

2

Calculate the fraction of the original panels that pass the second stage without defects or flaws.

\frac{11}{15}

of the original panels

Of the remaining

\frac{11}{12}

of the panels,

20\%

have cosmetic flaws, meaning

100\% - 20\% = 80\%

(or

\frac{4}{5}

) do not have flaws. Multiplying these fractions gives

\frac{4}{5} \times \frac{11}{12} = \frac{11}{15}

.

3

Calculate the fraction of original panels that pass the first two stages but are not selected for testing.

\frac{44}{75}

of the original panels

Since

\frac{1}{5}

of the flawless panels are selected for testing, the remaining

1 - \frac{1}{5} = \frac{4}{5}

are not selected. Multiplying this by the fraction from Step 2 gives

\frac{4}{5} \times \frac{11}{15} = \frac{44}{75}

.

4

Solve for the original number of panels,

P

, using the given value of

1,760

.

P = 3,000

panels

Set up the equation

\frac{44}{75} P = 1,760

. Solving for

P

gives

P = 1,760 \times \frac{75}{44} = 40 \times 75 = 3,000

.

Key Concept

Applying successive fractions and percentages to resolve a multi-stage word problem by working backward to the initial value.

Question 458Question →

What is the exact value of $\cos\left(\arcsin\left(\frac{3}{5}\right)\right)$ expressed as a decimal?

Show answer & explanation

Answer: 0.8

Answer

The exact value of the expression is 0.8.

Evaluating

\cos\left(\arcsin\left(\frac{3}{5}\right)\right)

requires finding the cosine of an angle

\theta

whose sine is

\frac{3}{5}

. In a right triangle with an opposite side of 3 and a hypotenuse of 5, the adjacent side is

\sqrt{5^2 - 3^2} = 4

. The cosine of

\theta

is the ratio of the adjacent side to the hypotenuse, which gives

\frac{4}{5} = 0.8

.

Step-by-Step Solution

1

Interpret the inverse sine function as an angle.

Let

\theta = \arcsin\left(\frac{3}{5}\right)

, meaning

\sin(\theta) = \frac{3}{5}

for

0 < \theta < \frac{\pi}{2}

.

The inverse sine function returns an angle whose sine is the given value within the principal interval

\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]

.

2

Find the adjacent side of the right triangle associated with angle

\theta

.

\text{adjacent} = \sqrt{5^2 - 3^2} = \sqrt{16} = 4

.

By the Pythagorean theorem,

a^2 + b^2 = c^2

, so the adjacent side length is

\sqrt{c^2 - b^2}

.

3

Calculate the cosine of angle

\theta

.

\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} = 0.8

.

Cosine is defined as the ratio of the adjacent side to the hypotenuse in a right triangle.

Key Concept

Composition of Trigonometric and Inverse Trigonometric Functions

Estimated Time:45s

Question 459Question →

In rectangle $ABCD$ , the length of side $AD$ is $12$ units and the length of side $CD$ is $17$ units. Point $E$ lies on side $CD$ such that $\triangle ADE$ is an isosceles right triangle with the right angle at vertex $D$ . What is the length, in units, of segment $BE$ ?

Show answer & explanation

Answer: 13

Answer

The length of segment

BE

is

13

units.

Because

\triangle ADE

is an isosceles right triangle with the right angle at vertex

D

, leg

DE

equals leg

AD = 12

. Subtracting

DE

from total side length

CD = 17

gives segment

EC = 5

. Since

ABCD

is a rectangle, angle

C

is a right angle (

90^\circ

) and

BC = AD = 12

. Applying the Pythagorean Theorem to right triangle

\triangle BCE

gives hypotenuse

BE = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13

.

Step-by-Step Solution

1

Find the length of segment

DE

using the properties of an isosceles right triangle.

DE = 12

units

In isosceles right triangle

\triangle ADE

with right angle at

D

, legs

AD

and

DE

are equal in length. Given

AD = 12

,

DE

must also be

12

.

2

Determine the length of segment

EC

.

EC = 5

units

Since point

E

lies on side

CD

,

EC = CD - DE = 17 - 12 = 5

.

3

Use the Pythagorean Theorem in right triangle

\triangle BCE

to find

BE

.

BE = 13

units

Because

ABCD

is a rectangle, angle

C

is

90^\circ

and

BC = AD = 12

. Applying the Pythagorean Theorem with legs

BC = 12

and

EC = 5

gives

BE = \sqrt{12^2 + 5^2} = \sqrt{169} = 13

.

Key Concept

Applying properties of isosceles right triangles (

45^\circ-45^\circ-90^\circ

) and the Pythagorean Theorem in composite figures.

Question 460Question →

If $\sin \theta - \cos \theta = 0.6$ , what is the value of $\sin \theta \cos \theta$ ?

Show answer & explanation

Answer: 0.32

Answer

The value of

\sin \theta \cos \theta

is 0.32.

Squaring both sides of

\sin \theta - \cos \cos \theta = 0.6

yields

\sin^2 \theta - 2\sin \theta \cos \theta + \cos^2 \theta = 0.36

. Substituting the Pythagorean identity

\sin^2 \theta + \cos^2 \theta = 1

gives

1 - 2\sin \theta \cos \theta = 0.36

, which rearranges to

2\sin \theta \cos \theta = 0.64

. Dividing by 2 yields

\sin \theta \cos \theta = 0.32

.

Step-by-Step Solution

1

Square both sides of the given equation

(\sin \theta - \cos \theta)^2 = 0.36

Squaring allows us to introduce the product

\sin \theta \cos \theta

alongside

\sin^2 \theta

and

\cos^2 \theta

.

2

Expand the binomial on the left side

\sin^2 \theta - 2\sin \theta \cos \theta + \cos^2 \theta = 0.36

Use the algebraic expansion identity

(a - b)^2 = a^2 - 2ab + b^2

.

3

Substitute the fundamental Pythagorean trigonometric identity

\sin^2 \theta + \cos^2 \theta = 1

1 - 2\sin \theta \cos \theta = 0.36

\sin^2 \theta + \cos^2 \theta

always equals 1 for any angle

\theta

.

4

Isolate the term containing

\sin \theta \cos \theta

2\sin \theta \cos \theta = 0.64

Subtract 0.36 from 1 to find the value of

2\sin \theta \cos \theta

.

5

Divide by 2 to solve for

\sin \theta \cos \theta

\sin \theta \cos \theta = 0.32

Simplifies

0.64 / 2

to obtain the final required numerical value.

Key Concept

Pythagorean identity

\sin^2 \theta + \cos^2 \theta = 1

Answer

Step-by-Step Solution

Key Concept

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