All practice questions

Question 541Question →

For an acute angle $x$ measured in degrees, $\sin(x) - \sin(90^\circ - x) = \frac{1}{5}$ . What is the value of $12(\tan(x) + \tan(90^\circ - x))$ ?

Show answer & explanation

Answer: 25

Answer

The value of the expression is 25.

Applying the co-function identity

\sin(90^\circ - x) = \cos(x)

allows the given equation to be written as

\sin(x) - \cos(x) = \frac{1}{5}

. Squaring both sides of this equation yields

\sin^2(x) - 2\sin(x)\cos(x) + \cos^2(x) = \frac{1}{25}

. Applying the Pythagorean identity

\sin^2(x) + \cos^2(x) = 1

simplifies this to

1 - 2\sin(x)\cos(x) = \frac{1}{25}

, which gives

\sin(x)\cos(x) = \frac{12}{25}

. The expression to be evaluated is

12(\tan(x) + \tan(90^\circ - x))

. Using the identity

\tan(90^\circ - x) = \cot(x)

, this expression can be rewritten as

12\left(\frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)}\right) = 12\left(\frac{\sin^2(x) + \cos^2(x)}{\sin(x)\cos(x)}\right) = 12\left(\frac{1}{\sin(x)\cos(x)}\right)

. Substituting the value of

\sin(x)\cos(x)

gives

12 \times \frac{25}{12} = 25

.

Step-by-Step Solution

1

Apply the co-function identity to rewrite the equation.

\sin(x) - \cos(x) = \frac{1}{5}

Since

\sin(90^\circ - x) = \cos(x)

for any angle

x

, we can substitute

\cos(x)

into the given equation.

2

Square both sides of the rewritten equation.

\sin^2(x) - 2\sin(x)\cos(x) + \cos^2(x) = \frac{1}{25}

Squaring both sides allows us to use the Pythagorean trigonometric identity to find the product of sine and cosine.

3

Substitute the Pythagorean identity and solve for

\sin(x)\cos(x)

.

\sin(x)\cos(x) = \frac{12}{25}

Substituting

\sin^2(x) + \cos^2(x) = 1

yields

1 - 2\sin(x)\cos(x) = \frac{1}{25}

, which simplifies to

\sin(x)\cos(x) = \frac{12}{25}

.

4

Rewrite the target expression in terms of sine and cosine.

\tan(x) + \tan(90^\circ - x) = \frac{1}{\sin(x)\cos(x)}

Using the co-function identity

\tan(90^\circ - x) = \cot(x)

and expressing the tangent and cotangent functions as ratios of sine and cosine yields

\frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)} = \frac{\sin^2(x) + \cos^2(x)}{\sin(x)\cos(x)} = \frac{1}{\sin(x)\cos(x)}

.

5

Substitute the value of

\sin(x)\cos(x)

and multiply by 12.

25

Substituting

\sin(x)\cos(x) = \frac{12}{25}

into

12(\tan(x) + \tan(90^\circ - x))

gives

12 \times \frac{25}{12} = 25

.

Key Concept

Applying co-function identities, the Pythagorean identity, and fundamental trigonometric relations to simplify expressions.

Estimated Time:2m 0s

Question 542Question →

A circle has a radius of $10$ . A shaded sector of the circle has an area of $10\pi$ . What is the measure, in degrees, of the central angle of the shaded sector?

Show answer & explanation

Answer: 36

Answer

The measure of the central angle of the shaded sector is 36 degrees.

The total area of the circle is

A = \pi (10)^2 = 100\pi

. The shaded sector's area is

10\pi

, which represents

\frac{10\pi}{100\pi} = \frac{1}{10}

of the total area of the circle. Since a circle consists of

360^\circ

, the central angle of the sector is

\frac{1}{10} \times 360^\circ = 36^\circ

.

Step-by-Step Solution

1

Calculate the total area of the circle.

Total area =

100\pi

The area of a circle is calculated using the formula

A = \pi r^2

where the radius

r = 10

.

2

Determine the proportion of the circle covered by the shaded sector.

Proportion =

\frac{1}{10}

Dividing the sector's area of

10\pi

by the total area of

100\pi

gives the fraction of the circle represented by the sector.

3

Find the central angle in degrees.

Central angle = 36

Since a full circle has a central angle of

360^\circ

, multiplying the proportion

\frac{1}{10}

by

360^\circ

yields the central angle of the sector.

Key Concept

The ratio of the area of a sector to the total area of a circle is equal to the ratio of the sector's central angle measure to the total degree measure of a circle (

360^\circ

).

Estimated Time:45s

Question 543Question →

A municipal utility department wants to estimate the number of residential water service lines that contain lead in a community with $3,600$ homes. The department selects a random sample of $120$ homes and inspects their water service lines. The inspection reveals that $9$ of the sampled homes have lead service lines. Based on the results of this sample, what is the estimated number of homes in the entire community that have lead service lines?

Show answer & explanation

Answer: 270

Answer

The estimated number of homes in the community that have lead service lines is 270.

Based on the random sample, the proportion of homes with lead service lines is 9 out of 120, which is 0.075 (or 7.5%). Generalizing this to the entire population of 3,600 homes yields an estimate of

0.075 \times 3,600 = 270

homes.

Step-by-Step Solution

1

Calculate the proportion of homes in the sample with lead service lines.

\frac{9}{120} = 0.075

To find the sample proportion, divide the number of homes with lead service lines by the total number of homes inspected.

2

Estimate the total number of homes in the community with lead service lines.

0.075 \times 3,600 = 270

Multiply the total number of homes in the community by the proportion found in the random sample.

Key Concept

Estimating population parameters and totals from a random sample.

Question 544Question →

A construction project requires a specific concrete mixture made by combining cement, sand, and gravel in a ratio of $1 : 2 : 4$ by weight. A builder needs to prepare exactly $350$ pounds of this concrete mixture. If the builder currently has $80$ pounds of sand and an unlimited supply of cement and gravel, how many additional pounds of sand must the builder purchase to make the concrete mixture?

Show answer & explanation

Answer: 20

Answer

The builder must purchase 20 additional pounds of sand.

The ratio of cement to sand to gravel is

1 : 2 : 4

, giving a total of

1 + 2 + 4 = 7

parts. Therefore, sand represents

\frac{2}{7}

of the total weight of the mixture. To find the total sand required for a

350

-pound mixture, we calculate

\frac{2}{7} \times 350 = 100

pounds. Subtracting the

80

pounds of sand already on hand, we find that the builder must purchase

100 - 80 = 20

additional pounds of sand.

Step-by-Step Solution

1

Determine the fraction of the concrete mixture that consists of sand using the given ratio of

1 : 2 : 4

by weight.

Sand constitutes

\frac{2}{7}

of the total weight.

The sum of the ratio parts is

1 + 2 + 4 = 7

parts, and sand represents

2

parts out of these

7

total parts.

2

Calculate the total weight of sand required for

350

pounds of the concrete mixture.

100

pounds of sand.

Multiplying the fraction of sand,

\frac{2}{7}

, by the total desired weight of the mixture,

350

pounds, gives the required weight of sand.

3

Calculate the additional amount of sand needed by subtracting the available sand from the total required sand.

20

pounds of sand.

Since the builder already has

80

pounds of sand, subtracting this from the required

100

pounds yields the weight of additional sand to purchase.

Key Concept

Part-to-whole ratios and proportion scaling

Question 545Question →

A polynomial $f(x)$ is defined by $f(x) = 2x^3 + hx^2 - 7x - 6$ , where $h$ is a constant. If the graph of $y = f(x)$ in the $xy$ -plane passes through the point $(2, 0)$ , what is the value of $h$ ?

Show answer & explanation

Answer: 1

Answer

The value of the constant

h

is 1.

Since the graph of

y = f(x)

passes through the point

(2, 0)

, the value of the function at

x = 2

must be

0

. Substituting

x = 2

into the equation yields

2(2)^3 + h(2)^2 - 7(2) - 6 = 0

. Simplifying the terms gives

2(8) + 4h - 14 - 6 = 0

, which simplifies to

16 + 4h - 20 = 0

, or

4h - 4 = 0

. Solving for

h

yields

h = 1

.

Step-by-Step Solution

1

Apply the point condition to the polynomial function.

f(2) = 0

Since the point

(2, 0)

lies on the graph of

y = f(x)

, substituting

x = 2

must yield

y = 0

.

2

Substitute

x = 2

into the expression for

f(x)

.

f(2) = 2(2)^3 + h(2)^2 - 7(2) - 6

We replace each occurrence of

x

with

2

to evaluate the function value.

3

Simplify the algebraic expression.

16 + 4h - 20 = 0 \implies 4h - 4 = 0

Evaluating the exponents and multiplications gives

2(8) + 4h - 14 - 6 = 16 + 4h - 20 = 4h - 4

.

4

Solve the linear equation for

h

.

h = 1

Adding 4 to both sides gives

4h = 4

, and dividing by 4 gives

h = 1

.

Key Concept

Connecting graphical x-intercepts of a polynomial to its algebraic roots and evaluation

Question 546Question →

In triangle $PQR$ , point $S$ lies on side $PQ$ and point $T$ lies on side $PR$ such that segment $ST$ is parallel to segment $QR$ . The length of segment $PS$ is $x$ , the length of segment $SQ$ is $6$ , the length of segment $ST$ is $x + 2$ , and the length of segment $QR$ is $2x + 7$ . What is the length of segment $QR$ ?

Show answer & explanation

Answer: 15

Answer

The length of segment

QR

is

15

.

Since segment

ST

is parallel to segment

QR

, triangle

PST

is similar to triangle

PQR

. The ratio of their corresponding sides is equal, so

\frac{PS}{PQ} = \frac{ST}{QR}

. Substituting the values gives

\frac{x}{x + 6} = \frac{x + 2}{2x + 7}

. Cross-multiplying and simplifying yields

x^2 - x - 12 = 0

. Factoring gives

(x-4)(x+3)=0

, so

x=4

because side lengths must be positive. Substituting

x=4

into

2x+7

yields

15

.

Step-by-Step Solution

1

Determine the similarity between the triangles

Triangle

PST

is similar to triangle

PQR

Since segment

ST

is parallel to segment

QR

, the corresponding angles are equal, establishing similarity by Angle-Angle (AA) criterion.

2

Express the total length of side

PQ

PQ = x + 6

The length of side

PQ

is the sum of the collinear segments

PS

and

SQ

.

3

Set up the similarity ratio equation

\frac{x}{x + 6} = \frac{x + 2}{2x + 7}

Corresponding side lengths of similar triangles are in proportion:

\frac{PS}{PQ} = \frac{ST}{QR}

.

4

Solve the quadratic equation for

x

x = 4

Cross-multiplying gives

x(2x + 7) = (x + 2)(x + 6) \implies 2x^2 + 7x = x^2 + 8x + 12 \implies x^2 - x - 12 = 0 \implies (x - 4)(x + 3) = 0

. Since length must be positive,

x = 4

.

5

Calculate the length of segment

QR

QR = 15

Substitute

x = 4

into the expression for

QR

, which is

2x + 7

, resulting in

2(4) + 7 = 15

.

Key Concept

Using triangle similarity and algebraic equations to find unknown segment lengths.

Question 547Question →

For an acute angle $\theta$ , the expression $\frac{\sin^4(\theta) - \cos^4(\theta)}{\sin(\theta) - \cos(\theta)}$ is equal to $\frac{7}{5}$ . What is the value of $25\sin(\theta)\cos(\theta)$ ?

Show answer & explanation

Answer: 12

Answer

The correct answer is 12.

The expression

\frac{\sin^4(\theta) - \cos^4(\theta)}{\sin(\theta) - \cos(\theta)}

simplifies directly to

\sin(\theta) + \cos(\theta) = \frac{7}{5}

by applying the difference of squares identity twice and substituting the Pythagorean identity

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

. Squaring both sides of this simplified equation gives

\sin^2(\theta) + 2\sin(\theta)\cos(\theta) + \cos^2(\theta) = \frac{49}{25}

. Replacing

\sin^2(\theta) + \cos^2(\theta)

with 1 yields

1 + 2\sin(\theta)\cos(\theta) = \frac{49}{25}

, which simplifies to

\sin(\theta)\cos(\theta) = \frac{12}{25}

. Multiplying this result by 25 gives the final integer value of 12.

Step-by-Step Solution

1

Factor the numerator of the expression.

\sin^4(\theta) - \cos^4(\theta) = (\sin^2(\theta) - \cos^2(\theta))(\sin^2(\theta) + \cos^2(\theta))

The difference of squares identity can be applied to terms with fourth powers.

2

Apply the Pythagorean identity to simplify the factored numerator.

\sin^4(\theta) - \cos^4(\theta) = \sin^2(\theta) - \cos^2(\theta)

The Pythagorean trigonometric identity states that

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

.

3

Factor the remaining term in the numerator.

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

This is another application of the difference of squares identity.

4

Simplify the fraction by dividing the common factor in the numerator and denominator.

\sin(\theta) + \cos(\theta) = \frac{7}{5}

The term

\sin(\theta) - \cos(\theta)

in the numerator and denominator cancels out since

\theta

is an acute angle and

\sin(\theta) \neq \cos(\theta)

.

5

Square both sides of the simplified equation.

\sin^2(\theta) + 2\sin(\theta)\cos(\theta) + \cos^2(\theta) = \frac{49}{25}

Squaring both sides allows us to relate the sum

\sin(\theta) + \cos(\theta)

to the product

\sin(\theta)\cos(\theta)

.

6

Substitute the Pythagorean identity and solve for the product of sine and cosine.

\sin(\theta)\cos(\theta) = \frac{12}{25}

Substituting

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

yields

1 + 2\sin(\theta)\cos(\theta) = \frac{49}{25}

, which simplifies to

2\sin(\theta)\cos(\theta) = \frac{24}{25}

.

7

Multiply the product by 25 to find the target value.

12

The question asks for the value of

25\sin(\theta)\cos(\theta)

.

Key Concept

Simplifying trigonometric expressions using algebraic factorization and Pythagorean identities.

Alternative Method

Since

\theta

is an acute angle in a right triangle, we can test standard Pythagorean triples. A right triangle with side lengths 3, 4, and 5 has an angle

\theta

where

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

and

\cos(\theta) = \frac{4}{5}

. Checking these values in the simplified expression gives

\sin(\theta) + \cos(\theta) = \frac{3}{5} + \frac{4}{5} = \frac{7}{5}

, which matches the given condition. We can then directly calculate

25\sin(\theta)\cos(\theta) = 25 \left(\frac{3}{5}\right)\left(\frac{4}{5}\right) = 12

.

Estimated Time:3m 0s

Question 548Question →

An agricultural department monitored the population of honeybee colonies in two adjacent conservation areas starting in 2018 ( $t = 0$ ). In Area 1, the number of colonies increases by a constant 230 colonies each year. In Area 2, the number of colonies increases by a constant 10% each year. At $t = 0$ , both areas had 2,000 colonies. If the model for Area 1 predicts $L$ colonies at $t = 3$ and the model for Area 2 predicts $E$ colonies at $t = 3$ , what is the value of $L - E$ ?

Show answer & explanation

Answer: 28

Answer

28

To find the value of

L - E

, the populations projected by each model at

t = 3

must be calculated.

For Area 1, the growth is linear with an initial population of 2,000 and a constant annual increase of 230. The model is

L(t) = 2000 + 230t

. At

t = 3

, the population is

L(3) = 2000 + 230(3) = 2690

.

For Area 2, the growth is exponential with an initial population of 2,000 and a constant annual increase of 10%. The model is

E(t) = 2000(1.10)^t

. At

t = 3

, the population is

E(3) = 2000(1.10)^3 = 2000(1.331) = 2662

.

Subtracting the exponential model output from the linear model output yields

L - E = 2690 - 2662 = 28

.

Step-by-Step Solution

1

Model the population growth for Area 1.

L(t) = 2000 + 230t

. For

t = 3

,

L(3) = 2000 + 230(3) = 2000 + 690 = 2690

. Thus,

L = 2690

.

Since Area 1 increases by a constant number of colonies (230) each year, its growth is linear. The initial population is 2,000.

2

Model the population growth for Area 2.

E(t) = 2000(1.10)^t

. For

t = 3

,

E(3) = 2000(1.10)^3 = 2000(1.331) = 2662

. Thus,

E = 2662

.

Since Area 2 increases by a constant percentage (10%) each year, its growth is exponential with a growth factor of

1 + 0.10 = 1.10

.

3

Calculate the difference

L - E

.

L - E = 2690 - 2662 = 28

.

We need to find the difference between the linear model's prediction and the exponential model's prediction at

t = 3

.

Key Concept

Linear growth increases by a constant amount per unit of time, whereas exponential growth increases by a constant percentage (or multiplies by a constant factor) per unit of time.

Alternative Method

Instead of setting up equations, the population values can be calculated step-by-step for each year.
Year 1 (

t = 1

):
Area 1:

2000 + 230 = 2230

Area 2:

2000 \times 1.10 = 2200

Year 2 (

t = 2

):
Area 1:

2230 + 230 = 2460

Area 2:

2200 \times 1.10 = 2420

Year 3 (

t = 3

):
Area 1:

2460 + 230 = 2690

Area 2:

2420 \times 1.10 = 2662

Subtracting the two values at Year 3 gives the final result:

2690 - 2662 = 28

.

Estimated Time:1m 30s

Question 549Question →

In a right triangle, the two acute angles are $\theta$ and $\phi$ . If $\sin(\theta) = \frac{7}{25}$ and $\cos(\phi) = \frac{k}{50}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 14

Answer

14

In any right triangle, the two acute angles,

\theta

and

\phi

, must sum to

90^\circ

. The cofunction trigonometric identity states that

\sin(\theta) = \cos(90^\circ - \theta) = \cos(\phi)

. Therefore, we can equate the two given values:

\frac{7}{25} = \frac{k}{50}

. Solving this equation for

k

gives

k = \frac{7 \times 50}{25} = 14

.

Step-by-Step Solution

1

Determine the relationship between the two acute angles in a right triangle.

\theta + \phi = 90^\circ

Since the sum of the angles in any triangle is

180^\circ

and a right triangle has one

90^\circ

angle, the sum of the other two acute angles must be

90^\circ

.

2

Apply the cofunction trigonometric identity for complementary angles.

\sin(\theta) = \cos(\phi)

The sine of an acute angle is always equal to the cosine of its complement.

3

Equate the given expressions and solve for

k

.

\frac{7}{25} = \frac{k}{50} \implies k = 14

Substitute the given values into the identity and multiply both sides by

50

to isolate

k

.

Key Concept

Cofunction identities for complementary angles in a right triangle

Question 550Question →

In a circle with center $O$ , central angle $AOB$ has a measure of $\frac{5\pi}{6}$ radians. If the radius of the circle is $12$ , the length of arc $AB$ is $k\pi$ . What is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

The value of

k

is

10

.

The arc length

s

subtended by a central angle

\theta

(in radians) in a circle of radius

r

is given by

s = r\theta

. Substituting the radius

r = 12

and the angle

\theta = \frac{5\pi}{6}

yields

s = 12 \times \frac{5\pi}{6} = 10\pi

. Since the arc length is expressed as

k\pi

, the value of

k

is

10

.

Step-by-Step Solution

1

Identify the formula for arc length in radians.

s = r\theta

The arc length

s

is directly proportional to the radius

r

and the central angle

\theta

in radians.

2

Substitute the given values into the formula.

s = 12 \times \frac{5\pi}{6}

The radius of the circle is

12

and the central angle is

\frac{5\pi}{6}

radians.

3

Calculate the arc length.

s = 10\pi

Simplifying

12 \times \frac{5}{6}

gives

2 \times 5 = 10

, so the product is

10\pi

.

4

Determine the value of

k

.

k = 10

We equate the calculated arc length

10\pi

with the given expression

k\pi

.

Key Concept

Arc length of a circle using radian measure

Estimated Time:45s

Question 551Question →

A research group conducted a survey using a random sample of $250$ residents of a town with a total population of $18,000$ residents. Of the residents surveyed, $64\%$ reported that they recycle regularly. The survey has a margin of error of $3\%$ . Based on the survey results, what is the difference between the maximum and minimum estimated number of residents in the town who recycle regularly?

Show answer & explanation

Answer: 1080

Answer

The correct answer is 1,080.

The sample proportion is

64\%

with a margin of error of

3\%

, which establishes an interval of

61\%

to

67\%

for the proportion of the population that recycles regularly. Multiplying these boundaries by the total population of

18,000

yields a minimum estimate of

0.61 \times 18,000 = 10,980

residents and a maximum estimate of

0.67 \times 18,000 = 12,060

residents. The difference between the maximum and minimum estimated number of residents is

12,060 - 10,980 = 1,080

. Alternatively, this difference can be found by multiplying the total width of the interval (

2 \times 0.03 = 0.06

) by the population size (

0.06 \times 18,000 = 1,080

).

Step-by-Step Solution

1

Determine the range of the proportion of residents who recycle regularly.

The proportion ranges from

61\%

(

0.61

) to

67\%

(

0.67

).

The margin of error of

3\%

is subtracted from and added to the sample proportion of

64\%

.

2

Calculate the minimum and maximum estimated number of residents who recycle regularly.

The minimum estimate is

10,980

residents and the maximum estimate is

12,060

residents.

Multiply the minimum and maximum proportions by the total population of

18,000

:

0.61 \times 18,000 = 10,980

and

0.67 \times 18,000 = 12,060

.

3

Find the difference between the maximum and minimum estimated values.

12,060 - 10,980 = 1,080

(or alternatively,

2 \times 0.03 \times 18,000 = 1,080

).

Subtracting the minimum estimate from the maximum estimate yields the difference.

Key Concept

Using sample statistics and margin of error to estimate population parameters.

Question 552Question →

For the polynomial $p(x) = 2x^3 - x^2 - kx + 6$ , where $k$ is a constant, the remainder when $p(x)$ is divided by $2x - 3$ is $0$ . What is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

The value of

k

is

7

.

The correct answer is

7

. According to the Factor Theorem, a linear expression

ax - b

is a factor of a polynomial

p(x)

if and only if

p\left(\frac{b}{a}\right) = 0

. Here, the divisor is

2x - 3

, so setting

2x - 3 = 0

gives the root

x = \frac{3}{2}

. Substituting

x = \frac{3}{2}

into

p(x) = 2x^3 - x^2 - kx + 6

gives

2\left(\frac{3}{2}\right)^3 - \left(\frac{3}{2}\right)^2 - k\left(\frac{3}{2}\right) + 6 = 0

. Simplifying this equation gives

\frac{27}{4} - \frac{9}{4} - \frac{3}{2}k + 6 = 0

, which reduces to

\frac{9}{2} - \frac{3}{2}k + 6 = 0

. Combining the constant terms gives

\frac{21}{2} - \frac{3}{2}k = 0

, which simplifies to

3k = 21

. Solving for

k

gives

k = 7

.

Step-by-Step Solution

1

Determine the root associated with the linear factor

2x - 3

.

Setting

2x - 3 = 0

gives

x = \frac{3}{2}

. By the Factor Theorem,

p\left(\frac{3}{2}\right) = 0

.

According to the Factor Theorem, a polynomial

p(x)

has a factor of the form

ax - b

if and only if

p\left(\frac{b}{a}\right) = 0

.

2

Substitute

x = \frac{3}{2}

into the polynomial expression

2x^3 - x^2 - kx + 6

and set it to

0

.

2\left(\frac{3}{2}\right)^3 - \left(\frac{3}{2}\right)^2 - k\left(\frac{3}{2}\right) + 6 = 0

This establishes a linear equation in terms of the unknown constant

k

.

3

Simplify the numerical terms and solve for

k

.

2\left(\frac{27}{8}\right) - \frac{9}{4} - \frac{3}{2}k + 6 = 0 \implies \frac{27}{4} - \frac{9}{4} - \frac{3}{2}k + 6 = 0 \implies \frac{9}{2} - \frac{3}{2}k + 6 = 0 \implies \frac{21}{2} - \frac{3}{2}k = 0 \implies 3k = 21 \implies k = 7

.

Standard fractional arithmetic and algebraic isolation are used to find the value of

k

.

Key Concept

Factor and Remainder Theorems

Question 553Question →

In the figure, lines $AB$ and $CD$ intersect at point $O$ , and ray $OE$ is perpendicular to line $AB$ . If the measure of $\angle EOC$ is $27^\circ$ , what is the measure, in degrees, of $\angle BOD$ ?

Show answer & explanation

Answer: 63

Answer

The measure of

\angle BOD

is 63 degrees.

The correct answer is 63. Since ray

OE

is perpendicular to line

AB

, the angle

\angle EOA

is a right angle measuring

90^\circ

. The adjacent angles

\angle AOC

and

\angle EOC

make up

\angle EOA

, which means

\angle AOC = 90^\circ - 27^\circ = 63^\circ

. Finally, because lines

AB

and

CD

intersect at point

O

, the angle

\angle BOD

and the angle

\angle AOC

are vertical angles. Vertical angles are equal in measure, so the measure of

\angle BOD

is

63^\circ

.

Step-by-Step Solution

1

Identify the angle formed by perpendicular lines.

The measure of

\angle EOA

is

90^\circ

.

Since ray

OE

is perpendicular to line

AB

, the angle

\angle EOA

is a right angle.

2

Calculate the measure of

\angle AOC

.

The measure of

\angle AOC

is

63^\circ

.

Angles

\angle AOC

and

\angle EOC

are adjacent and form the right angle

\angle EOA

, meaning they are complementary:

\angle AOC = 90^\circ - 27^\circ = 63^\circ

.

3

Determine the measure of

\angle BOD

.

The measure of

\angle BOD

is

63^\circ

.

Lines

AB

and

CD

intersect at point

O

, making

\angle BOD

and

\angle AOC

vertical angles. Since vertical angles are equal, the measure of

\angle BOD

is equal to the measure of

\angle AOC

.

Key Concept

Using properties of perpendicular lines and vertical angles to solve for unknown angle measures.

Estimated Time:1m 30s

Question 554Question →

In triangle $ABC$ , the measure of angle $ABC$ is $90^\circ$ and the measure of angle $BAC$ is $30^\circ$ . Segment $BD$ is perpendicular to segment $AC$ such that $D$ lies on $AC$ . Segment $DE$ is perpendicular to segment $BC$ such that $E$ lies on $BC$ . If the length of segment $CE$ is $3$ , what is the length of segment $AC$ ?

Show answer & explanation

Answer: 24

Answer

The correct answer is 24.

The correct answer is 24. Since the question asks for the length of segment AC, and by sequentially applying the properties of 30-60-90 right triangles we find CD = 6, BC = 12, and AC = 24.

Step-by-Step Solution

1

Find the measure of angle

ACB

in triangle

ABC

.

\angle ACB = 60^\circ

The sum of the angles in a triangle is

180^\circ

, so

\angle ACB = 180^\circ - \angle ABC - \angle BAC = 180^\circ - 90^\circ - 30^\circ = 60^\circ

.

2

Calculate the length of segment

CD

using right triangle

DEC

.

CD = 6

In the right triangle

DEC

,

\angle DEC = 90^\circ

and

\angle C = 60^\circ

, making it a 30-60-90 triangle. The side opposite the

30^\circ

angle is

CE = 3

, so the hypotenuse

CD

is

2 \times CE = 2 \times 3 = 6

.

3

Calculate the length of segment

BC

using right triangle

BDC

.

BC = 12

In the right triangle

BDC

,

BD

is perpendicular to

AC

, so

\angle BDC = 90^\circ

. With

\angle BCD = 60^\circ

, this is a 30-60-90 triangle. The side opposite the

30^\circ

angle is

CD = 6

, so the hypotenuse

BC

is

2 \times CD = 2 \times 6 = 12

.

4

Calculate the length of the hypotenuse

AC

using right triangle

ABC

.

AC = 24

In the right triangle

ABC

, the angle

\angle BAC = 30^\circ

and the side opposite to it is

BC = 12

. The hypotenuse

AC

is twice the length of the opposite leg, so

AC = 2 \times BC = 2 \times 12 = 24

.

Key Concept

Properties of special right triangles (30-60-90 triangles) and their trigonometric ratios.

Question 555Question →

At the beginning of 2015, Forest A had 4,000 trees and Forest B had 3,000 trees. The number of trees in Forest A increases by 150 trees each year. The number of trees in Forest B increases by 4% each year. If $A(t)$ and $B(t)$ represent the number of trees in Forest A and Forest B, respectively, $t$ years after the beginning of 2015, what is the value of $A(5) - B(5)$ , rounded to the nearest whole number?

Show answer & explanation

Answer: 1100

Answer

The correct value of

A(5) - B(5)

is 1100.

The correct value is 1100. By defining Forest A's population with the linear function

A(t) = 4000 + 150t

and Forest B's population with the exponential function

B(t) = 3000(1.04)^t

, evaluating both functions at

t = 5

yields

A(5) = 4750

and

B(5) \approx 3650

. The difference is

4750 - 3650 = 1100

.

Step-by-Step Solution

1

Formulate the growth model for Forest A.

A(t) = 4000 + 150t

Forest A grows linearly with a constant increase of 150 trees per year from an initial value of 4,000 trees.

2

Formulate the growth model for Forest B.

B(t) = 3000(1.04)^t

Forest B grows exponentially with a constant percent increase of 4% per year (growth factor of

1 + 0.04 = 1.04

) from an initial value of 3,000 trees.

3

Calculate the population of each forest after 5 years (

t = 5

).

A(5) = 4750

and

B(5) \approx 3650

Substitute

t = 5

into both models:

A(5) = 4000 + 150(5) = 4750

and

B(5) = 3000(1.04)^5 \approx 3649.96

.

4

Calculate the difference

A(5) - B(5)

and round to the nearest whole number.

1100

Subtract:

4750 - 3649.96 = 1100.04

. Rounding 1100.04 to the nearest whole number yields 1100.

Key Concept

Linear vs. Exponential Growth Models

Alternative Method

Instead of evaluating each function separately, you can construct the difference expression directly as

D(t) = (4000 + 150t) - 3000(1.04)^t

and substitute

t = 5

into

D(t)

to compute

D(5) = 4750 - 3000(1.04)^5 \approx 1100.04

, which rounds to 1100.

Estimated Time:1m 30s

Question 556Question →

An agricultural irrigation system uses two types of sprinklers, Type A and Type B. Water flows through $3$ Type A sprinklers at a combined rate of $15$ gallons per minute. Water flows through $5$ Type B sprinklers at a combined rate of $18$ gallons per minute. If a farmer runs $8$ Type A sprinklers and $10$ Type B sprinklers simultaneously, what is the total rate of water flow, in gallons per minute?

Show answer & explanation

Answer: 76

Answer

The total rate of water flow is 76 gallons per minute.

To find the total water flow rate, determine the rate of a single sprinkler of each type first. For Type A, the rate is

15 \text{ gallons per minute} \div 3 = 5 \text{ gallons per minute}

. For Type B, the rate is

18 \text{ gallons per minute} \div 5 = 3.6 \text{ gallons per minute}

. Running

8

Type A sprinklers and

10

Type B sprinklers simultaneously yields a total rate of

(8 \times 5) + (10 \times 3.6) = 40 + 36 = 76

gallons per minute.

Step-by-Step Solution

1

Find the rate of water flow for a single Type A sprinkler.

Each Type A sprinkler has a water flow rate of

5

gallons per minute.

Dividing the combined rate of

15

gallons per minute by the

3

sprinklers gives the individual rate.

2

Find the rate of water flow for a single Type B sprinkler.

Each Type B sprinkler has a water flow rate of

3.6

gallons per minute.

Dividing the combined rate of

18

gallons per minute by the

5

sprinklers gives the individual rate.

3

Calculate the total water flow rate for the combined group of active sprinklers.

The total combined flow rate is

76

gallons per minute.

Multiplying the individual rate of each type by the respective count of active sprinklers and summing the results:

(8 \times 5) + (10 \times 3.6) = 40 + 36 = 76

.

Key Concept

Combining rates of individual components to find a total rate.

Estimated Time:1m 30s

Question 557Question →

In the $xy$ -plane, an angle $\theta$ in standard position has its terminal ray intersecting the unit circle at the point $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$ . The angle is rotated counterclockwise by $15^\circ$ , and then its measure is doubled. The final resulting angle is coterminal with an angle of $\frac{a\pi}{b}$ radians, where $0 \le \frac{a\pi}{b} < 2\pi$ , and $a$ and $b$ are positive integers with no common factors. What is the value of $a + b$ ?

Show answer & explanation

Answer: 5

Answer

5

The correct answer is 5. Starting with the point

\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)

on the unit circle, the angle in standard position is

225^\circ

(or

\frac{5\pi}{4}

radians). Adding

15^\circ

for the counterclockwise rotation yields

240^\circ

(or

\frac{4\pi}{3}

radians). Doubling this measure gives

480^\circ

(or

\frac{8\pi}{3}

radians). Finding the coterminal angle within

[0, 2\pi)

yields

120^\circ

(or

\frac{2\pi}{3}

radians). In this form,

a=2

and

b=3

, which are positive integers with no common factors, so

a+b = 5

.

Step-by-Step Solution

1

Determine the initial angle of the terminal ray intersecting the unit circle at

\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)

.

The initial angle

\theta

is

225^\circ

(or

\frac{5\pi}{4}

radians).

Since both the

x

- and

y

-coordinates are negative and equal, the angle lies in the third quadrant and forms a

45^\circ

reference angle with the negative

x

-axis, which corresponds to

225^\circ

.

2

Apply a counterclockwise rotation of

15^\circ

to the initial angle.

The new angle is

240^\circ

(or

\frac{4\pi}{3}

radians).

Counterclockwise rotations correspond to adding positive angle measures:

225^\circ + 15^\circ = 240^\circ

.

3

Double the measure of the rotated angle.

The doubled angle is

480^\circ

(or

\frac{8\pi}{3}

radians).

The prompt specifies that the angle's measure is doubled after the rotation:

240^\circ \times 2 = 480^\circ

.

4

Find the coterminal angle within the standard interval

[0, 2\pi)

radians (or

[0^\circ, 360^\circ)

).

The coterminal angle is

120^\circ

(or

\frac{2\pi}{3}

radians).

To bring the angle back into the interval

[0^\circ, 360^\circ)

, subtract

360^\circ

(one full rotation):

480^\circ - 360^\circ = 120^\circ

. In radians,

120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3}

radians.

5

Identify

a

and

b

and calculate their sum.

a = 2

,

b = 3

, and

a + b = 5

.

The coterminal angle is expressed as

\frac{a\pi}{b}

in simplest form, so

a=2

and

b=3

, which are positive integers with no common factors.

Key Concept

Converting between radians and degrees, finding coterminal angles, and performing angle transformations.

Estimated Time:3m 0s

Question 558Question →

In the $xy$ -plane, the graph of the quadratic function $f$ has a vertex at $(h, k)$ , where $h$ and $k$ are constants. The graph passes through the points $(1, 10)$ and $(7, 10)$ . If the minimum value of the function $f$ is $1$ , what is the value of $f(2)$ ?

Show answer & explanation

Answer: 5

Answer

The value of

f(2)

is 5.

The correct answer is 5. Since the graph of the quadratic function passes through

(1, 10)

and

(7, 10)

, the axis of symmetry is the vertical line halfway between

x = 1

and

x = 7

, which is

x = \frac{1+7}{2} = 4

. The minimum value of the function is

1

, which occurs at the vertex, so the vertex is

(4, 1)

. In vertex form, the function is

f(x) = a(x-4)^2 + 1

. Substituting the point

(1, 10)

yields

10 = a(1-4)^2 + 1

, which simplifies to

9 = 9a

, so

a = 1

. The function is

f(x) = (x-4)^2 + 1

. Evaluating this function at

x = 2

gives

f(2) = (2-4)^2 + 1 = 5

.

Step-by-Step Solution

1

Find the

x

-coordinate of the vertex using the symmetry of the parabola.

h = 4

A parabola is symmetric about its vertical axis of symmetry. Since the points

(1, 10)

and

(7, 10)

have the same

y

-coordinate, the axis of symmetry is exactly halfway between their

x

-coordinates:

x = \frac{1 + 7}{2} = 4

.

2

Find the vertex coordinates using the minimum value of the function.

Vertex is

(4, 1)

The vertex of a quadratic function with a minimum value lies on the axis of symmetry, and its

y

-coordinate is the minimum value. Thus, the vertex

(h, k)

is

(4, 1)

.

3

Write the vertex form of the quadratic function and solve for the leading coefficient

a

.

f(x) = (x - 4)^2 + 1

Using the vertex form

f(x) = a(x - h)^2 + k

, we substitute

h = 4

and

k = 1

to get

f(x) = a(x - 4)^2 + 1

. Substituting the point

(1, 10)

gives

10 = a(1 - 4)^2 + 1 \implies 9 = 9a \implies a = 1

.

4

Evaluate

f(2)

using the determined function.

f(2) = 5

Substitute

x = 2

into the equation

f(x) = (x - 4)^2 + 1

to obtain

f(2) = (2 - 4)^2 + 1 = 4 + 1 = 5

.

Key Concept

Using symmetry and the vertex form of a quadratic function to determine its equation and evaluate values.

Estimated Time:1m 30s

Question 559Question →

A marine biologist wants to estimate the number of blue crabs in a bay that are infected with a specific parasite. The biologist captures a random sample of $250$ blue crabs from the bay, finds that $15$ of the crabs are infected, and then releases them. Based on this sample, if there is a total population of $12,000$ blue crabs in the bay, what is the best estimate of the total number of blue crabs infected with the parasite?

Show answer & explanation

Answer: 720

Answer

The best estimate of the total number of blue crabs infected with the parasite is 720.

The correct answer is found by establishing the sample proportion of infected crabs, which is 15 out of 250, or 6%. Multiplying this rate by the total estimated population of 12,000 crabs yields 720.

Step-by-Step Solution

1

Calculate the proportion of infected crabs in the sample.

\frac{15}{250} = 0.06

To determine the rate of infection in the representative sample.

2

Multiply the sample proportion by the total population of blue crabs in the bay.

0.06 \times 12,000 = 720

To generalize the sample rate to the entire population of crabs in the bay.

Key Concept

Generalizing a sample proportion to estimate a population parameter.

Question 560Question →

In the $xy$ -plane, a circle with center $(4, 9)$ and radius $5$ is defined by the equation $(x - h)^2 + (y - 9)^2 = 25$ , where $h$ is a positive constant. What is the value of $h$ ?

Show answer & explanation

Answer: 4

Answer

The value of the constant

h

is

4

.

The standard equation of a circle with center

(h, k)

and radius

r

is

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

. Since the center is

(4, 9)

and the radius is

5

, the equation of the circle is

(x - 4)^2 + (y - 9)^2 = 5^2

, which simplifies to

(x - 4)^2 + (y - 9)^2 = 25

. Comparing this to the given equation

(x - h)^2 + (y - 9)^2 = 25

, we see that the constant

h

corresponds to the

x

-coordinate of the center, which is

4

.

Step-by-Step Solution

1

Identify the standard equation of a circle.

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

This formula represents a circle with center

(h, k)

and radius

r

in the coordinate plane.

2

Substitute the given center coordinates into the standard equation.

For center

(4, 9)

, the equation is

(x - 4)^2 + (y - 9)^2 = r^2

.

The coordinates of the center are mapped directly to

h

and

k

in the standard form.

3

Compare the equation template to find the value of

h

.

The equation

(x - 4)^2 + (y - 9)^2 = 25

matches the format

(x - h)^2 + (y - 9)^2 = 25

, indicating that

h = 4

.

Equating the terms in both equations allows us to identify the value of the positive constant

h

.

Key Concept

Extracting coordinates of the center from the standard form equation of a circle.

Answer

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