All practice questions

Question 521Question →

A municipal solar power facility distributes its total generating capacity among three sectors: residential, commercial, and municipal. The residential sector is allocated $\frac{7}{15}$ of the total capacity, and the commercial sector is allocated $36\%$ of the total capacity. The remaining $26$ megawatts (MW) of capacity is allocated to municipal buildings. What is the total generating capacity, in megawatts, of the facility?

Show answer & explanation

Answer: 150

Answer

The total generating capacity of the facility is 150 megawatts.

The residential portion (

\frac{7}{15}

) and commercial portion (

36\% = \frac{9}{25}

) combine to equal

\frac{62}{75}

of the total capacity. The municipal portion makes up the remaining

\frac{13}{75}

of the total, which is given as

26

MW. Setting

\frac{13}{75} T = 26

and solving yields

T = 150

MW.

Step-by-Step Solution

1

Convert the commercial percentage into a simplified fraction.

36\% = \frac{36}{100} = \frac{9}{25}

Converting all proportions to fractions allows for direct operations.

2

Find the combined fraction of capacity allocated to residential and commercial sectors.

\frac{7}{15} + \frac{9}{25} = \frac{35}{75} + \frac{27}{75} = \frac{62}{75}

Finding a common denominator (

75

) allows adding the two fractional parts.

3

Find the fraction corresponding to the remaining municipal portion.

1 - \frac{62}{75} = \frac{13}{75}

Subtracting the allocated fraction from

1

yields the unallocated fractional portion.

4

Calculate the total capacity by setting up a linear equation.

\frac{13}{75} \times \text{Total} = 26 \implies \text{Total} = 26 \times \frac{75}{13} = 150

Multiplying the known remaining value by the reciprocal of its fraction gives the whole total.

Key Concept

Combining fractions and percentages to find an unknown total amount.

Estimated Time:1m 30s

Question 522Question →

An engineering formula used to calculate a structural load index is given by $L = \frac{a^3 b^{-2} - c^{3/4}}{a^2 + 12b}$ . What is the value of $L$ when $a = -2$ , $b = \frac{1}{3}$ , and $c = 16$ ?

Show answer & explanation

Answer: -10

Answer

The value of the expression is -10.

Substituting the values into the formula gives

L = \frac{(-2)^3 \cdot (1/3)^{-2} - 16^{3/4}}{(-2)^2 + 12(1/3)} = \frac{-8 \cdot 9 - 8}{4 + 4} = \frac{-72 - 8}{8} = \frac{-80}{8} = -10

.

Step-by-Step Solution

1

Evaluate the terms in the numerator containing powers and negative exponents

a^3 = -8

,

b^{-2} = 9

, and

c^{3/4} = 8

Negative bases raised to odd powers remain negative:

(-2)^3 = -8

. A negative exponent represents the reciprocal raised to a positive exponent:

(1/3)^{-2} = 3^2 = 9

. Fractional exponent

c^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8

.

2

Compute the full numerator

-72 - 8 = -80

Multiply

a^3

and

b^{-2}

to get

(-8)(9) = -72

, then subtract

c^{3/4} = 8

.

3

Evaluate the denominator

(-2)^2 + 12\left(\frac{1}{3}\right) = 4 + 4 = 8

Squaring a negative number yields a positive value:

(-2)^2 = 4

. Multiplying

12 \cdot \frac{1}{3} = 4

.

4

Divide the numerator by the denominator

\frac{-80}{8} = -10

Dividing a negative integer by a positive integer yields a negative result.

Key Concept

Evaluating algebraic expressions involving negative numbers, negative exponents, and rational exponents

Estimated Time:1m 15s

Question 523Question →

A box contains $10$ tiles labeled with the integers from $1$ through $10$ . If $2$ tiles are drawn at random without replacement, how many distinct pairs of tiles (where the order of selection does not matter) have a sum that is an odd number?

Show answer & explanation

Answer: 25

Answer

The total number of distinct pairs with an odd sum is 25.

To obtain an odd sum when adding two integers, one integer must be odd and the other must be even. In the range

1

through

10

, there are

5

odd integers (

1, 3, 5, 7, 9

) and

5

even integers (

2, 4, 6, 8, 10

). To form a pair with an odd sum, one tile must be selected from the

5

odd tiles and one tile must be selected from the

5

even tiles. By the Fundamental Counting Principle, the number of such distinct pairs is

5 \times 5 = 25

.

Step-by-Step Solution

1

Determine the parity condition for an odd sum

One tile must be odd and the other must be even

The sum of two integers is odd if and only if one addend is odd and the other addend is even.

2

Count the number of odd and even options

5 odd tiles and 5 even tiles

Among the integers

1

through

10

, the odd numbers are

1, 3, 5, 7, 9

(

5

total) and the even numbers are

2, 4, 6, 8, 10

(

5

total).

3

Apply the Fundamental Counting Principle

5 × 5 = 25 distinct pairs

Selecting one odd tile out of 5 possibilities and one even tile out of 5 possibilities gives

5 \times 5 = 25

distinct unordered pairs.

Key Concept

Fundamental Counting Principle and Parity of Integers

Estimated Time:1m 0s

Question 524Question →

A regional sports club assigns identification codes to all of its members. Each code consists of $1$ letter chosen from the set $\{K, L, M, N\}$ , followed by $2$ digits chosen from $\{1, 2, 3, 4, 5\}$ such that no digit is repeated within a code, followed by $1$ symbol chosen from $\{*, \#\}$ . How many unique identification codes can be created using this system?

Show answer & explanation

Answer: 160

Answer

The total number of unique identification codes that can be created is

160

.

According to the Fundamental Counting Principle, to find the total number of multi-stage outcomes, multiply the number of choices at each stage. For the letter slot, there are

4

choices. For the two digit slots without repetition, there are

5 \times 4 = 20

choices. For the symbol slot, there are

2

choices. Multiplying these gives

4 \times 20 \times 2 = 160

unique identification codes.

Step-by-Step Solution

1

Determine the number of available letter choices for the first slot.

4 options

The set of allowed letters

\{K, L, M, N\}

contains 4 distinct elements.

2

Calculate the number of permutations for the two-digit section without repetition.

20 options

Choosing 2 distinct digits from 5 options gives

5 \times 4 = 20

possible outcomes.

3

Determine the number of available symbol choices for the last slot.

2 options

The set of allowed symbols

\{*, \#\}

contains 2 elements.

4

Multiply the number of choices for each slot using the Fundamental Counting Principle.

160 unique codes

Total codes =

4 \times 20 \times 2 = 160

.

Key Concept

Fundamental Counting Principle and Permutations without Repetition

Estimated Time:1m 0s

Question 525Question →

In July, a company's IT department logged a set of service tickets. Of these tickets, $\frac{3}{10}$ were categorized as hardware issues, $45\%$ were categorized as software issues, and the remaining 60 tickets were categorized as network issues. What was the total number of service tickets logged by the IT department in July?

Show answer & explanation

Answer: 240

Answer

The total number of service tickets logged in July was 240.

Converting

\frac{3}{10}

gives

30\%

. Combining hardware (

30\%

) and software (

45\%

) issues yields

75\%

of the total tickets. The network tickets make up the remaining

25\%

(

100\% - 75\%

). Setting

25\%

of the total equal to 60 tickets gives a total of

\frac{60}{0.25} = 240

service tickets.

Step-by-Step Solution

1

Convert the fraction of hardware tickets to a percentage

\frac{3}{10} = 0.30 = 30\%

Expressing all portions as percentages enables direct combination.

2

Sum the percentages for hardware and software tickets

30\% + 45\% = 75\%

This determines the combined percentage of the non-network tickets.

3

Determine the remaining percentage corresponding to network tickets

100\% - 75\% = 25\%

The total of all categories must sum to 100%.

4

Calculate the total number of service tickets

\frac{60}{0.25} = 240

Dividing the quantity of network tickets by their decimal equivalent of 0.25 gives the total count.

Key Concept

Combining fractions and percentages to solve for an unknown whole amount

Estimated Time:1m 30s

Question 526Question →

A fair spinner is divided into $8$ congruent sectors numbered $1$ through $8$ . A player spins the spinner twice in succession. How many of the $64$ possible outcomes result in a sum of the two spins that is strictly greater than $12$ ?

Show answer & explanation

Answer: 10

Answer

There are 10 outcomes that yield a sum strictly greater than 12.

Systematically listing all ordered pairs

(x, y)

from

\{1, 2, \dots, 8\} \times \{1, 2, \dots, 8\}

such that

x + y > 12

yields 4 outcomes for sum 13, 3 outcomes for sum 14, 2 outcomes for sum 15, and 1 outcome for sum 16, totaling 10 valid outcomes.

Step-by-Step Solution

1

Determine the acceptable sums for the two spins

The possible sums strictly greater than 12 are 13, 14, 15, and 16.

Since each spin has a maximum value of 8, the maximum possible sum is 8 + 8 = 16.

2

Count the ordered pairs (spin 1, spin 2) for each valid sum

4 outcomes for sum 13, 3 outcomes for sum 14, 2 outcomes for sum 15, and 1 outcome for sum 16.

First and second spins are ordered, so (5,8) and (8,5) represent distinct outcomes.

3

Sum the outcome counts across all valid cases

4 + 3 + 2 + 1 = 10 outcomes.

The sets of outcomes for distinct sums are mutually exclusive.

Key Concept

Basic Probability and Counting Sample Space Outcomes

Estimated Time:1m 15s

Question 527Question →

Biophysicists conducted an experiment measuring the action potential conduction velocity ( $v$ , in $\text{m/s}$ ) in unmyelinated giant nerve fibers as a function of fiber diameter ( $d$ , in $\mu\text{m}$ ) at a constant temperature of $20^\circ\text{C}$ . The recorded data is presented in Table 1.

Fiber Diameter ( $d$ , $\mu\text{m}$ )	Conduction Velocity ( $v$ , $\text{m/s}$ )
100	5.0
200	8.0
400	14.0
600	18.0
900	21.0

Based on Table 1, what is the average rate of change in conduction velocity, in $\text{m/s}$ per $100\,\mu\text{m}$ increase in fiber diameter, over the interval from $d = 200\,\mu\text{m}$ to $d = 600\,\mu\text{m}$ ?

Show answer & explanation

Answer: 2.5

Answer

The average rate of change in conduction velocity over the specified interval is

2.5\,\text{m/s per }100\,\mu\text{m}

increase in fiber diameter.

To find the average rate of change in conduction velocity per

100\,\mu\text{m}

increase in fiber diameter between

d = 200\,\mu\text{m}

and

d = 600\,\mu\text{m}

, subtract the initial velocity (

8.0\,\text{m/s}

) from the final velocity (

18.0\,\text{m/s}

) to get

\Delta v = 10.0\,\text{m/s}

. Divide by the change in diameter

\Delta d = 600 - 200 = 400\,\mu\text{m}

to obtain

0.025\,\text{m/s per }\mu\text{m}

. Multiplying by

100

yields

2.5\,\text{m/s per }100\,\mu\text{m}

.

Step-by-Step Solution

1

Locate data points for

d = 200\,\mu\text{m}

and

d = 600\,\mu\text{m}

in Table 1.

At

d = 200\,\mu\text{m}

,

v = 8.0\,\text{m/s}

. At

d = 600\,\mu\text{m}

,

v = 18.0\,\text{m/s}

.

These data points define the boundaries of the interval specified in the question.

2

Calculate the overall changes in velocity (

\Delta v

) and diameter (

\Delta d

).

\Delta v = 18.0 - 8.0 = 10.0\,\text{m/s}

and

\Delta d = 600 - 200 = 400\,\mu\text{m}

.

Calculating the rate of change requires dividing the change in the dependent variable by the change in the independent variable.

3

Scale the rate of change to a

100\,\mu\text{m}

diameter interval.

\frac{10.0\,\text{m/s}}{400\,\mu\text{m}} \times 100\,\mu\text{m} = 2.5\,\text{m/s per }100\,\mu\text{m}

.

The question asks specifically for the rate per

100\,\mu\text{m}

increase in diameter.

Key Concept

Calculating average rate of change and trend slopes from quantitative scientific data tables.

Question 528Question →

A school library receives a shipment of $12$ new books, consisting of $5$ science fiction novels, $4$ historical fiction novels, and $3$ biography books. A librarian wants to choose $3$ books to feature on a display shelf. How many different combinations of $3$ books can be formed that contain exactly $2$ science fiction novels and $1$ biography book?

Show answer & explanation

Answer: 30

Answer

The total number of combinations containing exactly 2 science fiction novels and 1 biography book is 30.

To find the total number of combinations, first determine the number of ways to choose

2

science fiction novels from

5

available:

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

. Next, determine the number of ways to choose

1

biography book from

3

available:

\binom{3}{1} = 3

. Finally, apply the Fundamental Counting Principle by multiplying the counts together:

10 \times 3 = 30

.

Step-by-Step Solution

1

Calculate the combinations for selecting the science fiction novels.

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

Select

2

novels out of the

5

available science fiction novels without regard to order.

2

Calculate the combinations for selecting the biography book.

\binom{3}{1} = 3

Select

1

book out of the

3

available biography books.

3

Apply the Fundamental Counting Principle to find total combinations.

10 \times 3 = 30

Multiply the combinations calculated for each independent selection.

Key Concept

Combinations and the Fundamental Counting Principle

Question 529Question →

An art gallery manager is arranging 6 unique paintings side by side in a single line on a display wall. Two specific paintings created by the same artist must be placed directly next to each other. How many different line arrangements of the 6 paintings are possible?

Show answer & explanation

Answer: 240

Answer

The total number of different line arrangements is 240.

To satisfy the constraint that two specific paintings must be adjacent, treat them as a single combined block. This reduces the problem to arranging 5 items (the single block plus the other 4 paintings), which can be ordered in 5! = 120 ways. Within the block, the two paintings can be ordered in 2! = 2 ways. Applying the Fundamental Counting Principle gives 120 * 2 = 240 distinct arrangements.

Step-by-Step Solution

1

Group the two specific paintings together as a single block.

There are 5 units to arrange (1 block of 2 paintings plus 4 individual paintings).

Since the two paintings must remain adjacent, treating them as a single item guarantees they will stay together.

2

Calculate the permutations of the 5 units.

5! = 5 * 4 * 3 * 2 * 1 = 120 ways.

There are 5 distinct units being arranged in a straight line.

3

Determine the internal arrangements of the grouped pair.

2! = 2 * 1 = 2 ways.

The two specific paintings within the block can switch positions (Painting A then B, or Painting B then A).

4

Multiply the arrangements together.

120 * 2 = 240 total arrangements.

By the Fundamental Counting Principle, total arrangements equal the number of ways to place the units multiplied by the internal arrangements of the restricted pair.

Key Concept

Permutations with Restrictions (Grouped Elements)

Estimated Time:1m 15s

Question 530Question →

A student is creating a flag design consisting of 3 vertical stripes in a row. The student can choose from 5 different colors: Red, Blue, Green, Yellow, and White. To ensure visual contrast, any 2 adjacent stripes must be different colors. How many different flag designs can be created under these rules?

Show answer & explanation

Answer: 80

Answer

80

To find the total number of flag designs, apply the Fundamental Counting Principle across the three sequential stripe choices. The first stripe can be any of the 5 colors. The second stripe cannot match the first stripe's color, leaving 4 options. The third stripe cannot match the second stripe's color, which also leaves 4 options (since it can re-use the first stripe's color). Multiplying these choices together gives

5 \times 4 \times 4 = 80

unique flag designs.

Step-by-Step Solution

1

Determine choices for the first stripe

5 choices

Any of the 5 available colors can be selected for the first stripe.

2

Determine choices for the second stripe

4 choices

The second stripe must differ in color from the first stripe.

3

Determine choices for the third stripe

4 choices

The third stripe must differ in color from the second stripe (it may be the same color as the first stripe).

4

Apply the Fundamental Counting Principle

5 \times 4 \times 4 = 80

Multiply the options available at each stage to calculate the total number of distinct flag designs.

Key Concept

Fundamental Counting Principle with adjacent restrictions

Estimated Time:1m 15s

Question 531Question →

If $x = -3$ , $y = -\frac{1}{2}$ , and $z = 8$ , what is the value of the algebraic expression $\frac{x^2 y^{-1} + z^{2/3}}{2xy - 1}$ ?

Show answer & explanation

Answer: -7

Answer

The value of the expression is -7.

Substituting the given values into the expression requires careful application of exponent rules and order of operations. First,

(-3)^2 = 9

and

(-\frac{1}{2})^{-1} = -2

, so

x^2 y^{-1} = 9 \times (-2) = -18

. Second,

8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4

. This makes the numerator

-18 + 4 = -14

. The denominator evaluates to

2(-3)(-\frac{1}{2}) - 1 = 3 - 1 = 2

. Dividing

-14

by

2

gives the correct answer of

-7

.

Step-by-Step Solution

1

Substitute

x = -3

and

y = -\frac{1}{2}

into

x^2 y^{-1}

(-3)^2 \cdot (-2) = -18

Squaring

-3

yields

9

, and taking the reciprocal of

-\frac{1}{2}

yields

-2

.

2

Substitute

z = 8

into

z^{2/3}

8^{2/3} = 4

Taking the cube root of

8

gives

2

, and squaring

2

gives

4

.

3

Evaluate the numerator

-18 + 4 = -14

Adding the evaluated terms together.

4

Substitute values into the denominator

2xy - 1

2(-3)\left(-\frac{1}{2}\right) - 1 = 3 - 1 = 2

Multiplying

2

,

-3

, and

-\frac{1}{2}

produces

3

, then subtracting

1

yields

2

.

5

Divide the numerator by the denominator

\frac{-14}{2} = -7

Simplifying the rational expression yields the final numeric answer.

Key Concept

Evaluating algebraic expressions with negative bases, rational exponents, and negative exponents

Question 532Question →

If $x = -3$ , $y = -\frac{1}{2}$ , and $z = 4$ , what is the numerical value of the algebraic expression $\frac{x^2 - 4y^2}{x + yz}$ ?

Show answer & explanation

Answer: -1.6

Answer

The numerical value of the expression is

-1.6

.

Substituting

x = -3

,

y = -\frac{1}{2}

, and

z = 4

into

\frac{x^2 - 4y^2}{x + yz}

gives a numerator of

(-3)^2 - 4\left(-\frac{1}{2}\right)^2 = 9 - 4\left(\frac{1}{4}\right) = 8

and a denominator of

-3 + \left(-\frac{1}{2}\right)(4) = -3 - 2 = -5

. Evaluating

\frac{8}{-5}

yields

-1.6

.

Step-by-Step Solution

1

Evaluate the numerator

x^2 - 4y^2

8

Squaring

-3

gives

9

, and squaring

-\frac{1}{2}

gives

\frac{1}{4}

. Thus,

9 - 4\left(\frac{1}{4}\right) = 9 - 1 = 8

.

2

Evaluate the denominator

x + yz

-5

Multiplying

-\frac{1}{2}

by

4

gives

-2

. Adding

-3 + (-2)

yields

-5

.

3

Compute the final fraction quotient

-1.6

Dividing the numerator

8

by the denominator

-5

yields

-1.6

.

Key Concept

Evaluating algebraic expressions using order of operations with negative bases and fractional values.

Estimated Time:1m 30s

Question 533Question →

The polynomial $4x^2 - 8x - 5$ is subtracted from the polynomial $7x^2 - 3x + 4$ . The simplified difference can be expressed as $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constant integers. What is the value of the coefficient $b$ ?

Show answer & explanation

Answer: 5

Answer

The coefficient of the

x

term,

b

, is 5.

Subtracting

4x^2 - 8x - 5

from

7x^2 - 3x + 4

yields

(7x^2 - 4x^2) + (-3x - (-8x)) + (4 - (-5)) = 3x^2 + 5x + 9

. The coefficient of the linear term

x

is

5

.

Step-by-Step Solution

1

Set up the subtraction expression.

(7x^2 - 3x + 4) - (4x^2 - 8x - 5)

Subtracting the second polynomial from the first requires enclosing the second polynomial in parentheses to apply the subtraction to all terms.

2

Distribute the negative sign to all terms inside the parentheses.

7x^2 - 3x + 4 - 4x^2 + 8x + 5

Distributing the subtraction sign flips the sign of each term: positive terms become negative, and negative terms become positive.

3

Group and combine like terms.

3x^2 + 5x + 9

Combine the coefficients of matching variable parts:

(7 - 4)x^2 = 3x^2

,

(-3 + 8)x = 5x

, and

4 - (-5) = 9

.

4

Identify the coefficient

b

corresponding to the

x

term.

b = 5

Comparing the simplified expression

3x^2 + 5x + 9

to

ax^2 + bx + c

shows that the coefficient of

x

is 5.

Key Concept

Polynomial Subtraction and Combining Like Terms

Question 534Question →

If $x$ is a real number such that $\log_5(x) + \log_5(x - 20) = 3$ , what is the value of $x$ ?

Show answer & explanation

Answer: 25

Answer

The value of

x

is 25.

Applying the logarithmic product rule simplifies the equation to

\log_5(x^2 - 20x) = 3

. Writing this in exponential form yields

x^2 - 20x = 125

. Rearranging into standard form gives

x^2 - 20x - 125 = 0

, which factors into

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

. This gives potential solutions of

25

and

-5

. Because the logarithmic arguments must be strictly positive,

x = -5

is extraneous. Therefore, the only correct value is 25.

Step-by-Step Solution

1

Use the product property of logarithms to combine the terms on the left side.

\log_5(x(x - 20)) = 3

The sum of logarithms with the same base is equal to the logarithm of the product of their arguments:

\log_b(M) + \log_b(N) = \log_b(MN)

.

2

Rewrite the logarithmic equation in exponential form.

x(x - 20) = 5^3 \implies x^2 - 20x = 125

The logarithmic equation

\log_b(y) = c

is equivalent to the exponential equation

b^c = y

.

3

Rearrange the quadratic equation into standard form and solve by factoring.

x^2 - 20x - 125 = 0 \implies (x - 25)(x + 5) = 0 \implies x = 25 \text{ or } x = -5

Subtracting 125 from both sides sets the quadratic equation to 0, which can then be factored into binomials whose product is 0.

4

Verify the potential solutions in the original equation to identify any extraneous roots.

For

x = -5

, the arguments of the original logarithms are negative, which is undefined. For

x = 25

, the arguments are positive. Thus, the only valid solution is

x = 25

.

Logarithmic functions are only defined for positive real numbers. Therefore, we must have

x > 0

and

x - 20 > 0

, which requires

x > 20

.

Key Concept

Solving logarithmic equations by combining logarithmic terms and checking for extraneous solutions.

Estimated Time:1m 30s

Question 535Question →

In $\triangle ABC$ , the measure of angle $A$ is $40^\circ$ . Point $D$ lies on side $AC$ such that segment $BD$ bisects angle $ABC$ . If the measure of angle $BDC$ is $75^\circ$ , what is the measure, in degrees, of angle $C$ ?

Show answer & explanation

Answer: 70

Answer

The measure of angle C is 70 degrees.

The correct measure of angle

C

is found by first identifying that angle

ADB

is supplementary to angle

BDC

, giving a measure of

105^\circ

. Using the triangle angle sum theorem on triangle

ABD

, we find that angle

ABD

is

35^\circ

. Since

BD

bisects angle

ABC

, angle

DBC

is also

35^\circ

. Finally, applying the triangle angle sum theorem to triangle

BCD

, we subtract the measures of angles

DBC

(

35^\circ

) and

BDC

(

75^\circ

) from

180^\circ

to get

70^\circ

.

Step-by-Step Solution

1

Find the measure of angle

ADB

using the supplementary angle relationship with angle

BDC

.

105^\circ

Angles

ADB

and

BDC

form a linear pair along the line segment

AC

, so their sum is

180^\circ

.

2

Find the measure of angle

ABD

using the sum of interior angles in

\triangle ABD

.

35^\circ

The sum of interior angles in any triangle is

180^\circ

. Therefore, the measure of angle

ABD

is

180^\circ - (40^\circ + 105^\circ) = 35^\circ

.

3

Find the measure of angle

DBC

using the definition of an angle bisector.

35^\circ

Since segment

BD

bisects angle

ABC

, the measures of angles

ABD

and

DBC

must be equal.

4

Find the measure of angle

C

using the sum of interior angles in

\triangle BCD

.

70^\circ

The sum of interior angles in

\triangle BCD

is

180^\circ

. Therefore, the measure of angle

C

is

180^\circ - (35^\circ + 75^\circ) = 70^\circ

.

Key Concept

Using the triangle angle sum theorem and angle bisector properties to determine unknown angle measures in a geometric figure.

Question 536Question →

In right triangle $ABC$ , the hypotenuse $AC$ has a length of 13 centimeters, and leg $AB$ has a length of 5 centimeters. What is the length, in centimeters, of leg $BC$ ?

Show answer & explanation

Answer: 12

Answer

The length of leg

BC

is 12 centimeters.

Applying the Pythagorean theorem, we have

5^2 + BC^2 = 13^2

, which simplifies to

25 + BC^2 = 169

. Subtracting 25 from both sides gives

BC^2 = 144

, and taking the square root of both sides gives

BC = 12

centimeters.

Step-by-Step Solution

1

Set up the Pythagorean Theorem equation for right triangle

ABC

.

AB^2 + BC^2 = AC^2

The Pythagorean Theorem states that in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse.

2

Substitute the given values for

AB

and

AC

into the formula.

5^2 + BC^2 = 13^2

The length of leg

AB

is given as 5 centimeters, and the length of the hypotenuse

AC

is given as 13 centimeters.

3

Solve for the unknown leg length

BC

.

BC = 12

Squaring the values gives

25 + BC^2 = 169

. Subtracting 25 from both sides yields

BC^2 = 144

. Taking the square root of both sides gives

BC = 12

.

Key Concept

Pythagorean Theorem

Alternative Method

Recognize the triangle as a standard 5-12-13 Pythagorean triple, which immediately gives the missing leg length of 12 without needing calculations.

Estimated Time:30s

Question 537Question →

The circle $x^2 + y^2 = 25$ and the line $y = 2x - 5$ intersect at two points. What is the sum of the $y$ -coordinates of these two intersection points?

Show answer & explanation

Answer: -2

Answer

The sum of the

y

-coordinates of the intersection points is

-2

.

Substituting

y = 2x - 5

into the circular equation

x^2 + y^2 = 25

yields the quadratic equation

5x^2 - 20x = 0

. Factoring this equation gives

x = 0

and

x = 4

. Evaluating the linear equation at these values gives the

y

-coordinates

-5

and

3

. The sum of these coordinates is

-5 + 3 = -2

.

Step-by-Step Solution

1

Substitute

y = 2x - 5

into the circle equation

x^2 + y^2 = 25

.

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

To find the points of intersection, we solve the system of equations by substitution.

2

Expand and simplify the resulting equation.

5x^2 - 20x = 0

Expanding

(2x - 5)^2

gives

4x^2 - 20x + 25

. Combining like terms and subtracting 25 from both sides simplifies the equation.

3

Factor the quadratic equation to solve for

x

.

x = 0

or

x = 4

Factoring out

5x

gives

5x(x - 4) = 0

, which yields the roots

x = 0

and

x = 4

.

4

Find the corresponding

y

-coordinates by substituting the

x

-values back into

y = 2x - 5

.

The intersection points are

(0, -5)

and

(4, 3)

.

For

x = 0

,

y = 2(0) - 5 = -5

. For

x = 4

,

y = 2(4) - 5 = 3

.

5

Calculate the sum of the

y

-coordinates.

-2

Adding the

y

-coordinates

-5

and

3

gives

-5 + 3 = -2

.

Key Concept

Systems of Linear and Non-Linear Equations

Question 538Question →

In the standard $(x, y)$ coordinate plane, a triangle has vertices at $A(0, 1)$ , $B(2, 3)$ , and $C(8, 11)$ . If point $M$ is the midpoint of side $AB$ and point $N$ is the midpoint of side $AC$ , what is the length of the line segment $MN$ ?

Show answer & explanation

Answer: 5

Answer

The length of the line segment

MN

is

5

.

The length of the line segment

MN

is

5

. Calculating the coordinates of the midpoint of

AB

, we get

M(1, 2)

. For

AC

, the midpoint is

N(4, 6)

. Applying the distance formula between

M

and

N

yields

\sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9+16} = 5

. Alternatively, by the Midsegment Theorem, the segment connecting the midpoints of two sides of a triangle is half the length of the third side. The length of the third side

BC

is

\sqrt{(8-2)^2 + (11-3)^2} = \sqrt{36 + 64} = 10

, so the length of

MN

is

\frac{10}{2} = 5

.

Step-by-Step Solution

1

Find the coordinates of

M

, the midpoint of side

AB

with endpoints

A(0, 1)

and

B(2, 3)

.

M(1, 2)

The midpoint formula is

M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

.

2

Find the coordinates of

N

, the midpoint of side

AC

with endpoints

A(0, 1)

and

C(8, 11)

.

N(4, 6)

Applying the midpoint formula gives

\left(\frac{0 + 8}{2}, \frac{1 + 11}{2}\right) = (4, 6)

.

3

Calculate the distance between

M(1, 2)

and

N(4, 6)

using the distance formula.

5

The distance formula is

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

. Here,

d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5

.

Key Concept

Midpoint and Distance Formulas

Question 539Question →

In a convex polygon, the measures of the interior angles form an arithmetic progression. The smallest interior angle measures $120^\circ$ , and the common difference between consecutive interior angles is $5^\circ$ . What is the number of sides of this polygon?

Show answer & explanation

Answer: 9

Answer

The number of sides of the polygon is 9.

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

. Since the angles form an arithmetic progression with the first term

a = 120^\circ

and common difference

d = 5^\circ

, their sum is also given by the arithmetic series formula:

S_n = \frac{n}{2}[2(120) + (n-1)5] = \frac{n}{2}(5n + 235)

. Setting the two sums equal yields

\frac{n}{2}(5n + 235) = 180(n - 2)

, which simplifies to the quadratic equation

n^2 - 25n + 144 = 0

. Solving this gives

n = 9

or

n = 16

. Because the polygon is convex, every interior angle must be less than

180^\circ

. If

n = 16

, the largest angle would be

120^\circ + 15(5^\circ) = 195^\circ

, which is impossible for a convex polygon. If

n = 9

, the largest angle is

120^\circ + 8(5^\circ) = 160^\circ

, which is valid. Therefore, the number of sides must be 9.

Step-by-Step Solution

1

Set up the equation equating the geometric sum of interior angles to the arithmetic series sum.

The sum of the interior angles of a convex

n

-gon is

(n - 2) \times 180^\circ

. The sum of the arithmetic sequence of angles is

S_n = \frac{n}{2}[2(120^\circ) + (n - 1)5^\circ]

. Setting them equal gives:

\frac{n}{2}(5n + 235) = 180(n - 2)

.

This establishes the algebraic relationship between the polygon's geometric properties and the given sequence of angle measures.

2

Simplify the equation and solve the resulting quadratic equation for

n

.

5n^2 + 235n = 360n - 720 \Rightarrow 5n^2 - 125n + 720 = 0 \Rightarrow n^2 - 25n + 144 = 0

. Factoring the quadratic yields

(n - 9)(n - 16) = 0

, so

n = 9

or

n = 16

.

Solving the quadratic equation yields all mathematically possible values for the number of sides.

3

Apply the convexity constraint to determine the valid number of sides.

For a polygon to be convex, every interior angle must be less than

180^\circ

. For

n = 16

, the largest angle is

120^\circ + 15(5^\circ) = 195^\circ

, which is impossible. For

n = 9

, the largest angle is

120^\circ + 8(5^\circ) = 160^\circ

, which is valid.

The definition of a convex polygon requires all interior angles to be strictly less than

180^\circ

, which eliminates the extraneous solution of 16.

Key Concept

Sum of interior angles of a convex polygon and arithmetic progressions

Question 540Question →

The equation $x^2 + y^2 - 12x + 8y + 3 = 0$ defines a circle in the standard $(x, y)$ coordinate plane. What is the radius of this circle?

Show answer & explanation

Answer: 7

Answer

The radius of the circle is 7.

Completing the square on the given equation yields

(x - 6)^2 + (y + 4)^2 = 49

. By comparing this to the standard circle equation

(x - h)^2 + (y - k)^2 = r^2

, we find

r^2 = 49

. Taking the square root of

49

gives a radius of

7

.

Step-by-Step Solution

1

Group the

x

and

y

terms and move the constant to the right-hand side of the equation.

x^2 - 12x + y^2 + 8y = -3

This groups terms containing the same variables together to prepare for completing the square.

2

Add the square of half of the linear coefficients to both sides to complete the square for both

x

and

y

.

(x^2 - 12x + 36) + (y^2 + 8y + 16) = -3 + 36 + 16

Adding

(\frac{-12}{2})^2 = 36

and

(\frac{8}{2})^2 = 16

to both sides forms perfect square trinomials on the left side while maintaining equality.

3

Factor the trinomials into squared binomials and combine the constants on the right side.

(x - 6)^2 + (y + 4)^2 = 49

This rewrites the equation in the standard circle equation form.

4

Extract the radius from the standard form equation

(x - h)^2 + (y - k)^2 = r^2

.

r = 7

Since the constant on the right side corresponds to

r^2

, taking the square root of

49

gives the radius of the circle.

Key Concept

Converting a circle's equation from general form to standard form by completing the square to determine its properties.

Answer

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