Tüm alıştırma soruları

Soru 21Soru →

A boutique perfume workshop creates a signature fragrance blend by combining two fragrance oils, Oil A and Oil B. Oil A costs $12 per ounce and contains 40% pure essential oil by volume. Oil B costs$ 20 per ounce and contains 80% pure essential oil by volume. The perfumer creates a 30-ounce batch of the signature blend at a total cost that averages exactly $15 per ounce. What is the total volume, in ounces, of pure essential oil contained in this 30-ounce blend?

Cevabı ve açıklamayı göster

Cevap: 16.5

Cevap

The total volume of pure essential oil contained in the 30-ounce blend is 16.5 ounces.

To find the total essential oil content, first set up a system of equations for the volumes of Oil A (

x

) and Oil B (

y

):

x + y = 30

and

12x + 20y = 450

. Solving this system yields

x = 18.75

ounces and

y = 11.25

ounces. Multiplying each by its respective essential oil concentration gives

0.40 \times 18.75 = 7.5

ounces from Oil A and

0.80 \times 11.25 = 9

ounces from Oil B, totaling 16.5 ounces of essential oil.

Adım Adım Çözüm

1

Set up linear equations representing the total volume and total cost of the mixture.

x + y = 30

and

12x + 20y = 450

, where

x

is ounces of Oil A and

y

is ounces of Oil B.

The total cost of the 30-ounce blend at

15 per ounce is

15 \times 30 = 450$ dollars.

2

Solve for the quantities of Oil A and Oil B used in the mixture.

x = 18.75

ounces of Oil A and

y = 11.25

ounces of Oil B.

Substituting

y = 30 - x

into

12x + 20(30 - x) = 450

yields

-8x = -150

, giving

x = 18.75

.

3

Compute the amount of pure essential oil contributed by each component and sum them.

0.40(18.75) + 0.80(11.25) = 7.5 + 9 = 16.5

ounces.

Oil A contains 40% essential oil by volume and Oil B contains 80% essential oil by volume.

Anahtar Kavram

Systems of Linear Equations and Mixture Modeling

Tahmini Süre:2m 0s

Soru 22Soru →

If

x

is a positive integer such that

\frac{\sqrt{54 \cdot 4^{2x+1} + 9 \cdot 4^{2x}}}{\sqrt{2^{x+3} + 2^x}} = 2560

what is the value of

x

?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

x

is

6

.

Factoring out common exponential terms inside both radicals yields

\sqrt{225 \cdot 2^{4x}} = 15 \cdot 2^{2x}

for the numerator and

\sqrt{9 \cdot 2^x} = 3 \cdot 2^{x/2}

for the denominator. Dividing these gives

5 \cdot 2^{3x/2}

. Equating this to

2560

results in

2^{3x/2} = 512 = 2^9

, which simplifies to

\frac{3x}{2} = 9

, or

x = 6

.

Adım Adım Çözüm

1

Simplify the numerator inside the radical expression.

\sqrt{54 \cdot 4^{2x+1} + 9 \cdot 4^{2x}} = 15 \cdot 2^{2x}

Rewrite

4^{2x+1}

as

4 \cdot 4^{2x}

. Then factor out

4^{2x}

:

54(4 \cdot 4^{2x}) + 9 \cdot 4^{2x} = (216 + 9)4^{2x} = 225 \cdot 4^{2x}

. Taking the square root gives

\sqrt{225} \cdot \sqrt{(2^2)^{2x}} = 15 \cdot 2^{2x}

.

2

Simplify the denominator inside the radical expression.

\sqrt{2^{x+3} + 2^x} = 3 \cdot 2^{x/2}

Rewrite

2^{x+3}

as

2^3 \cdot 2^x = 8 \cdot 2^x

. Factoring out

2^x

gives

(8 + 1)2^x = 9 \cdot 2^x

. Taking the square root gives

\sqrt{9} \cdot \sqrt{2^x} = 3 \cdot 2^{x/2}

.

3

Simplify the quotient of the two radical expressions.

\frac{15 \cdot 2^{2x}}{3 \cdot 2^{x/2}} = 5 \cdot 2^{3x/2}

Divide the constants

\frac{15}{3} = 5

and subtract exponents with the same base:

2x - \frac{x}{2} = \frac{3x}{2}

.

4

Equate to 2560 and solve for

x

.

x = 6

Divide both sides by 5:

2^{3x/2} = \frac{2560}{5} = 512

. Express 512 as a power of 2:

512 = 2^9

. Therefore,

\frac{3x}{2} = 9 \implies 3x = 18 \implies x = 6

.

Anahtar Kavram

Exponent and Radical Simplification using Base Prime Factorization

Tahmini Süre:2m 0s

Soru 23Soru →

A quality control inspector evaluates a batch of $16$ precision components. Exactly $10$ of the components meet all engineering specifications, while $6$ have minor surface defects. If the inspector randomly selects $2$ components from the batch one after another without replacement, what is the probability that both selected components meet all engineering specifications?

Cevabı ve açıklamayı göster

Cevap: 0.375

Cevap

The probability that both selected components meet all engineering specifications is

0.375

(or

\frac{3}{8}

).

Since the components are selected without replacement, the outcome of the second draw depends on the outcome of the first draw. The probability of selecting a qualifying component first is

\frac{10}{16}

. Given that a qualifying component was drawn first,

9

qualifying components remain out of

15

total components. The joint probability of both events occurring is

\frac{10}{16} \times \frac{9}{15} = \frac{90}{240} = \frac{3}{8} = 0.375

.

Adım Adım Çözüm

1

Determine the probability of selecting a component meeting specifications on the first draw.

P(E_1) = \frac{10}{16} = \frac{5}{8}

There are

10

qualifying components out of

16

total components.

2

Determine the conditional probability of selecting a second component meeting specifications, given that the first component selected also met specifications.

P(E_2 \mid E_1) = \frac{9}{15} = \frac{3}{5}

Because sampling is done without replacement,

9

qualifying components remain out of

15

total remaining components.

3

Apply the multiplication rule for dependent events to calculate the probability of both events occurring.

P(E_1 \cap E_2) = P(E_1) \times P(E_2 \mid E_1) = \frac{5}{8} \times \frac{3}{5} = \frac{3}{8} = 0.375

For dependent events, the joint probability is the product of the first event's probability and the conditional probability of the second event.

Anahtar Kavram

Probability of Dependent Events without Replacement

Tahmini Süre:1m 15s

Soru 24Soru →

What is the smallest positive integer $n$ such that $n$ is a multiple of $180$ , the only prime factors of $n$ are $2$ , $3$ , and $5$ , and $n$ has exactly $36$ positive integer divisors?

Cevabı ve açıklamayı göster

Cevap: 1440

Cevap

1440

To minimize

n = 2^a \cdot 3^b \cdot 5^c

subject to

(a+1)(b+1)(c+1) = 36

with

a \ge 2

,

b \ge 2

, and

c \ge 1

, we evaluate all valid factor partitions of

36

. The partition

(a+1, b+1, c+1) = (6, 3, 2)

yields

(a, b, c) = (5, 2, 1)

, giving

n = 2^5 \cdot 3^2 \cdot 5^1 = 1440

, which is the smallest possible integer satisfying all criteria.

Adım Adım Çözüm

1

Express

n

in terms of prime factorization and establish exponent inequalities.

n = 2^a \cdot 3^b \cdot 5^c

with

a \ge 2

,

b \ge 2

, and

c \ge 1

.

Since

n

is a multiple of

180 = 2^2 \cdot 3^2 \cdot 5^1

and contains no other prime factors, its prime powers must at least match those of

180

.

2

Set up the divisor count equation.

(a+1)(b+1)(c+1) = 36

, where

a+1 \ge 3

,

b+1 \ge 3

, and

c+1 \ge 2

.

The total number of positive divisors of

2^a \cdot 3^b \cdot 5^c

is given by

(a+1)(b+1)(c+1)

.

3

Determine all valid factorizations of

36

into three factors

(x, y, z) = (a+1, b+1, c+1)

.

The valid factor sets

\{x, y, z\}

satisfying

x, y \ge 3

and

z \ge 2

are

\{4, 3, 3\}

and

\{6, 3, 2\}

.

Factor sets containing a factor of

2

for

x

or

y

(such as

\{9, 2, 2\}

) are invalid because

a+1 \ge 3

and

b+1 \ge 3

.

4

Calculate the value of

n

for all valid assignments of exponents.

From

\{4, 3, 3\}

:

(3, 2, 2) \implies 1800

,

(2, 3, 2) \implies 2700

,

(2, 2, 3) \implies 4500

.
From

\{6, 3, 2\}

:

(5, 2, 1) \implies 1440

,

(2, 5, 1) \implies 4860

.

Assigning larger exponents to smaller prime bases minimizes the overall product.

5

Select the minimum integer value among all candidates.

The smallest value is

1440

.

Comparing all valid candidates

1440, 1800, 2700, 4500, 4860

, the minimum is

1440

.

Anahtar Kavram

Prime Factorization and Divisor Count Constraints

Tahmini Süre:2m 30s

Soru 25Soru →

A chemical storage vessel contains a mixture composed of three liquid components: Component $X$ , Component $Y$ , and Component $Z$ . Initially, Component $X$ accounts for $\frac{3}{8}$ of the total mixture volume, and Component $Y$ accounts for $\frac{5}{12}$ of the total mixture volume, with Component $Z$ occupying the remainder of the volume.

During a processing stage, $\frac{1}{3}$ of Component $X$ is extracted and $\frac{2}{5}$ of Component $Y$ is extracted, while Component $Z$ remains completely unchanged. If the total volume of the mixture remaining in the vessel after processing is 170 milliliters, what was the total volume, in milliliters, of the mixture in the vessel before processing?

Cevabı ve açıklamayı göster

Cevap: 240

Cevap

The initial total volume of the mixture was 240 milliliters.

To find the initial total volume, first find the fraction of the initial mixture that is Component Z:

1 - (3/8 + 5/12) = 5/24

. Next, compute the remaining amounts of each component relative to the initial total volume

V

: Component X has

(1 - 1/3) \times 3/8 = 1/4 = 6/24

remaining; Component Y has

(1 - 2/5) \times 5/12 = 1/4 = 6/24

remaining; Component Z retains its

5/24

. Summing these yields

6/24 + 6/24 + 5/24 = 17/24

of the original volume. Setting

(17/24)V = 170

mL gives

V = 170 \times (24/17) = 240

mL.

Adım Adım Çözüm

1

Find the initial fraction of the mixture representing Component

Z

.

Component

Z = 1 - \left(\frac{3}{8} + \frac{5}{12}\right) = 1 - \left(\frac{9}{24} + \frac{10}{24}\right) = 1 - \frac{19}{24} = \frac{5}{24}

.

The sum of all component fractions must equal 1.

2

Calculate the remaining fraction for each component relative to the initial total volume

V

.

Remaining

X = \left(1 - \frac{1}{3}\right) \times \frac{3}{8}V = \frac{2}{3} \times \frac{3}{8}V = \frac{1}{4}V = \frac{6}{24}V

.
Remaining

Y = \left(1 - \frac{2}{5}\right) \times \frac{5}{12}V = \frac{3}{5} \times \frac{5}{12}V = \frac{1}{4}V = \frac{6}{24}V

.
Remaining

Z = \frac{5}{24}V

.

When a fraction of a component is removed, the remaining fraction of that component is multiplied by its original portion of the total mixture.

3

Sum the remaining component fractions to find the total remaining volume as a fraction of

V

.

Total Remaining Fraction

= \frac{6}{24}V + \frac{6}{24}V + \frac{5}{24}V = \frac{17}{24}V

.

Combining the remaining amounts gives the fraction of the initial mixture left.

4

Solve for the initial total volume

V

using the given final volume of 170 mL.

\frac{17}{24}V = 170 \implies V = 170 \times \frac{24}{17} = 10 \times 24 = 240 \text{ mL}.

Multiplying the final volume by the reciprocal of the remaining fraction yields the original volume.

Anahtar Kavram

Multi-step fraction operations and solving for original whole amounts

Soru 26Soru →

When the positive integer $n$ is divided by $9$ , the remainder is $5$ . What is the remainder when $n^2 + 4n + 7$ is divided by $9$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The remainder when

n^2 + 4n + 7

is divided by

9

is

7

.

Any positive integer

n

that leaves a remainder of

5

when divided by

9

can be represented as

n = 9k + 5

for some non-negative integer

k

. Substituting this into

n^2 + 4n + 7

yields

(9k + 5)^2 + 4(9k + 5) + 7 = 81k^2 + 90k + 25 + 36k + 20 + 7 = 81k^2 + 126k + 52

. Since

81k^2

and

126k

are both divisible by

9

, the remainder of the entire expression when divided by

9

is determined entirely by

52

. Dividing

52

by

9

gives

52 = 9 \times 5 + 7

, so the final remainder is

7

.

Adım Adım Çözüm

1

Express the integer

n

in terms of its remainder upon division by

9

.

n = 9k + 5

for some non-negative integer

k

, or equivalently

n \equiv 5 \pmod{9}

.

By the division algorithm, any integer divided by

9

can be written as a multiple of

9

plus the remainder.

2

Substitute

n \equiv 5 \pmod{9}

into the polynomial expression

n^2 + 4n + 7

.

n^2 + 4n + 7 \equiv 5^2 + 4(5) + 7 = 25 + 20 + 7 = 52 \pmod{9}

.

Properties of modular arithmetic allow substitution of remainder values into polynomial expressions.

3

Find the remainder of

52

when divided by

9

.

52 = 9 \times 5 + 7

, which gives a remainder of

7

.

Dividing

52

by

9

yields a quotient of

5

and a remainder of

7

, which is strictly between

0

and

8

.

Anahtar Kavram

Properties of Integer Remainders and Modular Substitution

Tahmini Süre:1m 15s

Soru 27Soru →

A sector of a circle with a radius of $12$ units has an area of $24\pi$ square units. What is the measure, in degrees, of the central angle of the sector?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The measure of the central angle of the sector is

60^\circ

.

The area of the full circle is

\pi r^2 = \pi (12)^2 = 144\pi

. The sector area represents a fraction of the total area, specifically

\frac{24\pi}{144\pi} = \frac{1}{6}

. Multiplying this fraction by the full circle's central angle of

360^\circ

gives

\frac{1}{6} \times 360^\circ = 60^\circ

.

Adım Adım Çözüm

1

Find the total area of the circle

The total area of the circle is

\pi \times 12^2 = 144\pi

.

The area of a full circle with radius

r

is given by

A = \pi r^2

.

2

Relate the sector area to the total circle area

The fraction of the circle represented by the sector is

\frac{24\pi}{144\pi} = \frac{1}{6}

.

The area of a sector is proportional to the fraction of the total central angle (

360^\circ

) it subtends.

3

Calculate the central angle in degrees

\theta = \frac{1}{6} \times 360^\circ = 60^\circ.

Multiply the fraction of the circle by

360^\circ

to get the central angle measure.

Anahtar Kavram

Relationship between central angle measure, total circle area, and sector area

Soru 28Soru →

If $2^{x + 3} \cdot 4^{x - 1} = 16^x$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The value of

x

is 1.

Rewriting

4^{x-1}

as

2^{2x-2}

and

16^x

as

2^{4x}

transforms the left side into

2^{x+3} \cdot 2^{2x-2} = 2^{3x+1}

. Setting exponents equal gives

3x + 1 = 4x

, which simplifies to

x = 1

.

Adım Adım Çözüm

1

Express all terms with a common base of 2

2^{x+3} \cdot (2^2)^{x-1} = (2^4)^x

Converting

4

to

2^2

and

16

to

2^4

allows all terms to share the base 2.

2

Apply the power of a power rule

(a^m)^n = a^{m \cdot n}

and the product rule

a^m \cdot a^n = a^{m+n}

2^{3x+1} = 2^{4x}

Multiplying exponents gives

(2^2)^{x-1} = 2^{2x-2}

and

(2^4)^x = 2^{4x}

. Adding exponents on the left gives

(x+3) + (2x-2) = 3x+1

.

3

Equate the exponents and solve for

x

x = 1

Since

2^A = 2^B

implies

A = B

, setting

3x + 1 = 4x

directly yields

x = 1

.

Anahtar Kavram

Solving exponential equations using common bases and exponent properties

Soru 29Soru →

The high temperatures, in degrees Fahrenheit, recorded in a city over a 7-day period were 64, 58, 75, 67, 61, 83, and 72. What is the interquartile range of these temperatures, in degrees Fahrenheit?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The interquartile range of the recorded temperatures is 14.

To calculate the interquartile range, first arrange the dataset in ascending order: 58, 61, 64, 67, 72, 75, 83. The median of the 7 values is 67. The first quartile (

Q_1

) is the median of the lower three values (58, 61, 64), which is 61. The third quartile (

Q_3

) is the median of the upper three values (72, 75, 83), which is 75. Subtracting

Q_1

from

Q_3

yields an interquartile range of

75 - 61 = 14

.

Adım Adım Çözüm

1

Order the dataset from least to greatest

58, 61, 64, 67, 72, 75, 83

Finding quartiles requires data to be arranged in ascending order.

2

Find the first quartile (

Q_1

) and third quartile (

Q_3

)

Q_1 = 61

and

Q_3 = 75

The median of the dataset (the 4th value) is 67. The lower half of the data consists of 58, 61, 64 (median 61), and the upper half consists of 72, 75, 83 (median 75).

3

Compute the difference between

Q_3

and

Q_1

75 - 61 = 14

The interquartile range is defined as

IQR = Q_3 - Q_1

.

Anahtar Kavram

Interquartile Range (IQR)

Soru 30Soru →

A triangle has a base of length $14$ centimeters and an area of $42$ square centimeters. What is the height, in centimeters, of the triangle perpendicular to this base?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The height of the triangle perpendicular to the base is 6 centimeters.

The area of a triangle is given by

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

. Substituting

42

for the area and

14

for the base yields

42 = \frac{1}{2} \times 14 \times h

, which simplifies to

42 = 7h

. Dividing

42

by

7

gives

h = 6

.

Adım Adım Çözüm

1

Write down the triangle area formula.

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

The area and base are known values, and the goal is to solve for the perpendicular height.

2

Substitute the given values into the formula.

42 = \frac{1}{2} \times 14 \times \text{height}

Plugging in

\text{Area} = 42

and

\text{base} = 14

creates a single-variable algebraic equation.

3

Solve for the height.

42 = 7 \times \text{height} \implies \text{height} = 6

Simplifying

\frac{1}{2} \times 14

to

7

and dividing both sides by

7

yields the height.

Anahtar Kavram

The area of a triangle is calculated using the formula Area = (1/2) * base * height.

Soru 31Soru →

If $x$ is an integer that satisfies both $|5 - 2x| \le 9$ and $|x + 1| > 3$ , what is the least possible value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The least possible value of

x

is 3.

Solving

|5 - 2x| \le 9

gives

-2 \le x \le 7

. Solving

|x + 1| > 3

gives

x > 2

or

x < -4

. Taking the intersection of both regions yields

2 < x \le 7

. The integer values satisfying this combined inequality are

3, 4, 5, 6,

and

7

. The smallest among these integer values is

3

.

Adım Adım Çözüm

1

Solve the first absolute value inequality

|5 - 2x| \le 9

-2 \le x \le 7

Remove the absolute value bars to set up the compound inequality

-9 \le 5 - 2x \le 9

. Subtracting

5

from all parts gives

-14 \le -2x \le 4

. Dividing all parts by

-2

and reversing the inequality signs yields

7 \ge x \ge -2

, or

-2 \le x \le 7

.

2

Solve the second absolute value inequality

|x + 1| > 3

x > 2

or

x < -4

Remove the absolute value bars to create two separate cases:

x + 1 > 3 \implies x > 2

, or

x + 1 < -3 \implies x < -4

.

3

Determine the set of values that satisfy both inequalities simultaneously

2 < x \le 7

The intersection of the interval

[-2, 7]

and

(-\infty, -4) \cup (2, \infty)

is

(2, 7]

, because

x < -4

does not overlap with

[-2, 7]

.

4

Find the smallest integer within the interval

(2, 7]

3

The integers included in the interval

(2, 7]

are

3, 4, 5, 6,

and

7

. Note that

2

is excluded due to the strict inequality

x > 2

. Therefore, the least possible integer value is

3

.

Anahtar Kavram

Linear Inequalities and Absolute Value

Soru 32Soru →

If $x$ is a real number that satisfies both $|3x - 12| \le 9$ and $|2 - x| \ge 4$ , the maximum possible value of the expression $5 - 2x$ is $M$ and the minimum possible value is $m$ . What is the value of $M - m$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

M - m

is 2.

Solving

|3x - 12| \le 9

yields

1 \le x \le 7

. Solving

|2 - x| \ge 4

yields

x \le -2

or

x \ge 6

. The set of

x

-values satisfying both conditions is the intersection

[6, 7]

. Since

5 - 2x

is a linear expression with a negative coefficient, its maximum value

M

occurs at the smallest value of

x

(

x = 6

), giving

M = 5 - 2(6) = -7

. Its minimum value

m

occurs at the largest value of

x

(

x = 7

), giving

m = 5 - 2(7) = -9

. Thus,

M - m = -7 - (-9) = 2

.

Adım Adım Çözüm

1

Solve the inequality

|3x - 12| \le 9

1 \le x \le 7

Expanding absolute value yields

-9 \le 3x - 12 \le 9

. Adding 12 gives

3 \le 3x \le 21

, then dividing by 3 yields

1 \le x \le 7

.

2

Solve the inequality

|2 - x| \ge 4

x \le -2

or

x \ge 6

Absolute value inequality

|u| \ge a

splits into

u \ge a

or

u \le -a

. Here

2 - x \ge 4 \implies x \le -2

, and

2 - x \le -4 \implies x \ge 6

.

3

Determine the overlapping domain for

x

6 \le x \le 7

Combining

1 \le x \le 7

with

x \le -2

or

x \ge 6

leaves only the interval

6 \le x \le 7

.

4

Evaluate maximum

M

and minimum

m

of

5 - 2x

on

6 \le x \le 7

M = -7

and

m = -9

Since

-2x

decreases as

x

increases, the maximum occurs at

x = 6

(

M = 5 - 12 = -7

) and the minimum occurs at

x = 7

(

m = 5 - 14 = -9

).

5

Calculate the difference

M - m

2

Subtracting

m

from

M

gives

-7 - (-9) = 2

.

Anahtar Kavram

Linear Inequalities and Absolute Value

Tahmini Süre:2m 0s

Soru 33Soru →

The table below shows the distribution of scores achieved by 20 students on a final exam:

Score	Frequency
60	3
70	5
80	6
90	4
100	2

What is the interquartile range (IQR) of the scores?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The interquartile range (IQR) of the scores is 20.

For a set of 20 ordered scores, the first quartile

Q_1

is the median of the first 10 scores (average of the 5th and 6th values), and the third quartile

Q_3

is the median of the last 10 scores (average of the 15th and 16th values). Using the cumulative frequencies, the 5th and 6th scores are both 70 (

Q_1 = 70

), and the 15th and 16th scores are both 90 (

Q_3 = 90

). Thus, the interquartile range is

IQR = Q_3 - Q_1 = 90 - 70 = 20

.

Adım Adım Çözüm

1

Determine the cumulative frequency to locate quartile positions.

Score 60 occupies positions 1 to 3; Score 70 occupies positions 4 to 8; Score 80 occupies positions 9 to 14; Score 90 occupies positions 15 to 18; Score 100 occupies positions 19 to 20.

Tracking data positions in a frequency distribution allows efficient determination of medians and quartiles without expanding the raw list.

2

Calculate the first quartile (

Q_1

).

Q_1 = \frac{70 + 70}{2} = 70

.

With 20 total values, the lower half consists of the first 10 values (positions 1 through 10). The median of these 10 values is the average of the 5th and 6th values, which are both 70.

3

Calculate the third quartile (

Q_3

).

Q_3 = \frac{90 + 90}{2} = 90

.

The upper half consists of the last 10 values (positions 11 through 20). The median of these 10 values is the average of the 15th and 16th values, which are both 90.

4

Subtract

Q_1

from

Q_3

to find the interquartile range.

IQR = Q_3 - Q_1 = 90 - 70 = 20

.

The interquartile range represents the spread of the middle 50% of the dataset.

Anahtar Kavram

Interquartile Range (IQR) from a Frequency Distribution

Tahmini Süre:1m 30s

Soru 34Soru →

Two water pumps, Pump A and Pump B, were used to drain a reservoir containing 12,000 gallons of water. Pump A operates at a constant rate that is 50 gallons per hour greater than the rate of Pump B. Pump A worked alone for 4 hours, after which both pumps worked together for another 8 hours to completely empty the reservoir. What is the pumping rate of Pump B, in gallons per hour?

Cevabı ve açıklamayı göster

Cevap: 570

Cevap

570

Let

r

represent the rate of Pump B in gallons per hour. Pump A's rate is

(r + 50)

gallons per hour. In the first 4 hours, Pump A drains

4(r + 50) = 4r + 200

gallons. In the next 8 hours, both pumps operate together at a combined rate of

(r + 50) + r = 2r + 50

gallons per hour, draining

8(2r + 50) = 16r + 400

gallons. Adding both quantities gives total volume drained:

(4r + 200) + (16r + 400) = 12,000

. Simplifying gives

20r + 600 = 12,000

, so

20r = 11,400

, which yields

r = 570

gallons per hour.

Adım Adım Çözüm

1

Define variables for the rate of each pump

Rate of Pump B =

r

gal/hr; Rate of Pump A =

r + 50

gal/hr

Establishing a single unknown variable allows setting up a one-variable linear equation.

2

Write expressions for water drained during each time period

Period 1 (Pump A alone for 4 hrs):

4(r + 50) = 4r + 200

; Period 2 (Both pumps for 8 hrs):

8(2r + 50) = 16r + 400

Work done equals rate multiplied by time for each phase of operation.

3

Sum the work done in both periods to equal total volume and solve for

r

20r + 600 = 12,000 \Rightarrow 20r = 11,400 \Rightarrow r = 570

Solving the linear equation yields the exact rate of Pump B.

Anahtar Kavram

Linear Equations in One Variable

Tahmini Süre:2m 0s

Soru 35Soru →

In a circle, an arc of length $4\pi$ corresponds to a central angle of $40^\circ$ . If the area of the sector formed by this central angle is $k\pi$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

The value of

k

is 36.

Using the arc length equation

4\pi = \frac{40}{360} \cdot 2\pi r

, we solve for the radius

r = 18

. Substituting

r = 18

into the sector area formula

A = \frac{40}{360} \cdot \pi (18)^2

results in

A = 36\pi

. Thus,

k = 36

.

Adım Adım Çözüm

1

Calculate the radius of the circle using the arc length formula

Radius

r = 18

Arc length is related to central angle and radius by

L = \frac{\theta}{360^\circ} \cdot 2\pi r

. Substituting

L = 4\pi

and

\theta = 40^\circ

gives

4\pi = \frac{1}{9} \cdot 2\pi r \implies r = 18

.

2

Calculate the area of the sector using the radius and central angle

Sector Area

A = 36\pi

Sector area is calculated using

A = \frac{\theta}{360^\circ} \cdot \pi r^2

. Substituting

\theta = 40^\circ

and

r = 18

gives

A = \frac{1}{9} \cdot \pi (18^2) = 36\pi

.

3

Extract the coefficient

k

from

k\pi

k = 36

Comparing

36\pi

to

k\pi

directly yields

k = 36

.

Anahtar Kavram

Relationship between central angle, arc length, radius, and sector area

Soru 36Soru →

In a circle centered at point $O$ , sector $OAB$ has a central angle of $60^\circ$ and a radius of $12$ . A smaller circle is inscribed inside sector $OAB$ such that it is tangent to radius $OA$ , radius $OB$ , and arc $AB$ . If the area of the region inside sector $OAB$ that lies outside the inscribed circle is expressed in the form $k\pi$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The correct value of

k

is 8.

By using the geometry of the

30^\circ

-

60^\circ

-

90^\circ

right triangle formed by the angle bisector and the radius of tangency, the radius of the inscribed circle is found to be

r = 4

. Subtracting its area (

16\pi

) from the sector's area (

24\pi

) gives

8\pi

, so

k = 8

.

Adım Adım Çözüm

1

Determine the relationship between the radius of the larger circle

R

and the radius of the inscribed circle

r

.

OP = 2r

and

R = 3r

.

The center

P

of the inscribed circle lies on the angle bisector of

\angle AOB = 60^\circ

, creating a

30^\circ

angle with radius

OA

. The perpendicular distance from

P

to radius

OA

is

r

, so

\sin(30^\circ) = \frac{r}{OP} = \frac{1}{2}

, giving

OP = 2r

. Since the inscribed circle touches arc

AB

,

OP + r = R

, so

3r = R

.

2

Calculate the radius

r

of the inscribed circle.

r = 4

.

Given

R = 12

, solving

3r = 12

yields

r = 4

.

3

Compute the area of sector

OAB

.

\text{Area}_{\text{sector}} = 24\pi

.

The formula for the area of a sector is

\frac{\theta}{360^\circ} \pi R^2

. Here,

\frac{60^\circ}{360^\circ} \pi (12^2) = \frac{1}{6} \times 144\pi = 24\pi

.

4

Compute the area of the inscribed circle.

\text{Area}_{\text{circle}} = 16\pi

.

The area of a circle with radius

r = 4

is

\pi r^2 = \pi (4^2) = 16\pi

.

5

Subtract the area of the inscribed circle from the area of sector

OAB

to find

k

.

k = 8

.

\text{Area}_{\text{region}} = 24\pi - 16\pi = 8\pi

, which means

k = 8

.

Anahtar Kavram

Inscribed shapes within sectors, arc length, and sector area relations

Soru 37Soru →

Two commercial printing presses, Press A and Press B, operate at constant rates to print a total order of $N$ pages. Press A prints at a constant rate of $x$ pages per minute, while Press B prints at a constant rate that is $25$ pages per minute faster than Press A. Press A begins printing alone. After $20$ minutes, Press B is turned on, and both presses work simultaneously for an additional $30$ minutes. At that point, Press A stops, and Press B works alone for $10$ final minutes to complete the order. If Press B printed exactly $\frac{6}{11}$ of the total number of pages in the order, what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The value of

x

is 50.

To find

x

, calculate the total time each press operated. Press A ran for 20 minutes alone plus 30 minutes with Press B, giving 50 minutes total at

x

pages per minute (

50x

pages). Press B ran for 30 minutes with Press A plus 10 minutes alone, giving 40 minutes total at

(x + 25)

pages per minute (

40x + 1000

pages). The total order size is

N = 90x + 1000

. Setting Press B's output equal to

\frac{6}{11}N

yields

40x + 1000 = \frac{6}{11}(90x + 1000)

. Multiplying both sides by 11 gives

440x + 11000 = 540x + 6000

, which simplifies to

100x = 5000

, giving

x = 50

.

Adım Adım Çözüm

1

Determine total operating times and express pages printed by each press in terms of

x

Press A printed

50x

pages; Press B printed

40(x + 25) = 40x + 1000

pages.

Press A operated for 20 minutes alone plus 30 minutes together (50 minutes total). Press B operated for 30 minutes together plus 10 minutes alone (40 minutes total).

2

Write the expression for total pages

N

N = 50x + (40x + 1000) = 90x + 1000

The total pages in the order is the sum of the pages printed by Press A and Press B.

3

Formulate the linear equation in one variable using the given ratio

40x + 1000 = \frac{6}{11}(90x + 1000)

Press B printed exactly

\frac{6}{11}

of the total pages

N

.

4

Solve the linear equation for

x

x = 50

Multiplying both sides by 11 clears the fraction to give

440x + 11000 = 540x + 6000

, which simplifies to

100x = 5000

.

Anahtar Kavram

Formulating and solving linear equations in one variable from multi-step rate and work scenarios.

Soru 38Soru →

What value of $x$ satisfies the exponential equation $25^x - 4 \cdot 5^{x+1} = 125$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct answer is

2

.

Rewriting

25^x

as

(5^x)^2

and

4 \cdot 5^{x+1}

as

20 \cdot 5^x

transforms the equation into

(5^x)^2 - 20(5^x) - 125 = 0

. Substituting

u = 5^x

produces

u^2 - 20u - 125 = 0

, which factors into

(u - 25)(u + 5) = 0

. Because

5^x

must be greater than zero for all real values of

x

,

u = -5

yields no valid real solution. Thus,

5^x = 25 = 5^2

, giving

x = 2

.

Adım Adım Çözüm

1

Convert exponential expressions to a common base of 5.

25^x = (5^2)^x = (5^x)^2

and

4 \cdot 5^{x+1} = 4 \cdot 5 \cdot 5^x = 20 \cdot 5^x

.

Applying exponent laws

a^{m+n} = a^m \cdot a^n

and

(a^m)^n = a^{mn}

expresses terms in quadratic form with respect to

5^x

.

2

Formulate and factor the quadratic equation in terms of

u = 5^x

.

u^2 - 20u - 125 = 0 \implies (u - 25)(u + 5) = 0

, yielding

u = 25

or

u = -5

.

The equation reduces to standard quadratic form, which factors easily.

3

Solve for

x

while rejecting non-viable real roots.

5^x = 25 = 5^2 \implies x = 2

.

5^x = -5

has no real solution.

An exponential function with a positive base produces strictly positive output values for all real domain inputs.

Anahtar Kavram

Solving exponential equations reducible to quadratic form using exponent laws

Soru 39Soru →

At a charity fundraising event, standard tickets were sold for $\$45$ each and VIP tickets were sold for $\$80$ each. The number of standard tickets sold was $12$ more than twice the number of VIP tickets sold. If the total revenue generated from standard tickets exceeded the total revenue from VIP tickets by $\$1,830$ , how many VIP tickets were sold?

Cevabı ve açıklamayı göster

Cevap: 129

Cevap

129 VIP tickets were sold.

Letting

v

represent the number of VIP tickets sold, the number of standard tickets sold is

2v + 12

. Expressing the revenue condition yields the linear equation

45(2v + 12) - 80v = 1830

. Distributing 45 gives

90v + 540 - 80v = 1830

. Combining like terms results in

10v + 540 = 1830

. Subtracting 540 from both sides gives

10v = 1290

, which yields

v = 129

.

Adım Adım Çözüm

1

Define the unknown variable

Let

v

be the number of VIP tickets sold.

The question asks for the number of VIP tickets, making

v

a direct choice for the variable.

2

Translate the relationship between ticket quantities into an algebraic expression

Standard tickets sold

= 2v + 12

'12 more than twice the number of VIP tickets' translates directly to

2v + 12

.

3

Formulate total revenue expressions and construct the single-variable linear equation

45(2v + 12) - 80v = 1830

Total standard revenue minus total VIP revenue equals the given excess of

\$1,830

.

4

Distribute and combine like terms to solve for

v

90v + 540 - 80v = 1830 \implies 10v + 540 = 1830 \implies 10v = 1290 \implies v = 129

Applying standard algebraic operations isolates

v

on one side of the equation.

Anahtar Kavram

Formulating and solving a linear equation in one variable from a multi-step word problem context

Tahmini Süre:2m 0s

Soru 40Soru →

If $x$ and $y$ are real numbers such that $|x - 4| \le 3$ and $|y + 2| \le 5$ , what is the maximum possible value of $|x - y|$ ?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The maximum possible value of

|x - y|

is 14.

Solving

|x - 4| \le 3

gives the closed interval

[1, 7]

for

x

. Solving

|y + 2| \le 5

gives the closed interval

[-7, 3]

for

y

. The maximum possible value of

|x - y|

is the maximum distance between a point in

[1, 7]

and a point in

[-7, 3]

, which is

7 - (-7) = 14

.

Adım Adım Çözüm

1

Determine the range of possible values for

x

.

1 \le x \le 7

The inequality

|x - 4| \le 3

represents all numbers within distance 3 of 4 on the number line.

2

Determine the range of possible values for

y

.

-7 \le y \le 3

The inequality

|y + 2| \le 5

represents all numbers within distance 5 of -2 on the number line.

3

Find the maximum distance between any point

x

in

[1, 7]

and any point

y

in

[-7, 3]

.

14

The maximum absolute difference

|x - y|

occurs between the upper endpoint of the

x

-interval (

7

) and the lower endpoint of the

y

-interval (

-7

).

Anahtar Kavram

Absolute value inequalities as distance intervals on the real number line and maximizing differences between bounded variables

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