Tüm alıştırma soruları

Soru 561Soru →

If $\sin(2x + 10)^\circ = \cos(3x - 5)^\circ$ , where the measures of both angles are in degrees and are acute, what is the value of $x$ ?

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Cevap: 17

Cevap

17

According to the co-function identities of trigonometry, the sine of an acute angle is equal to the cosine of its complementary angle. Therefore, if

\sin(2x + 10)^\circ = \cos(3x - 5)^\circ

, the sum of the two angles must be

90^\circ

. This gives the equation

(2x + 10) + (3x - 5) = 90

. Simplifying the equation yields

5x + 5 = 90

. Subtracting 5 from both sides gives

5x = 85

, and dividing by 5 yields

x = 17

.

Adım Adım Çözüm

1

Relate the sine and cosine functions using the co-function identity.

Since

\sin(A) = \cos(B)

for acute angles, the angles must be complementary, so

A + B = 90^\circ

.

The co-function identity states that the sine of an angle is equal to the cosine of its complement.

2

Set up the algebraic equation using the given angle expressions.

(2x + 10) + (3x - 5) = 90

This expresses the condition that the sum of the two acute angles is equal to

90^\circ

.

3

Solve the equation for

x

.

5x + 5 = 90 \implies 5x = 85 \implies x = 17

Combine like terms and isolate

x

by performing basic arithmetic operations.

Anahtar Kavram

Co-function identities relate trigonometric functions of complementary angles, specifically

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

.

Soru 562Soru →

In the diagram shown, point $D$ lies on side $AB$ of triangle $ABC$ , and point $E$ lies on side $AC$ . The lengths of the segments are $AB = 20$ , $AC = 16$ , $AD = 8$ , and $AE = 10$ . If the area of quadrilateral $BCED$ is $54$ , what is the area of triangle $ADE$ ?

(Note: Figure not drawn to scale.)

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18

Triangles ADE and ACB share angle A. The ratios of the adjacent sides of angle A are AD/AC = 8/16 = 1/2 and AE/AB = 10/20 = 1/2. By the Side-Angle-Side (SAS) similarity theorem, triangle ADE is similar to triangle ACB. The ratio of their areas is equal to the square of their similarity ratio: (1/2)^2 = 1/4. Therefore, the area of triangle ACB is 4 times the area of triangle ADE. The area of quadrilateral BCED is the difference between the area of triangle ACB and the area of triangle ADE, which is 4 * Area(ADE) - Area(ADE) = 3 * Area(ADE). Since the area of quadrilateral BCED is 54, we have 3 * Area(ADE) = 54, which simplifies to Area(ADE) = 18.

Adım Adım Çözüm

1

Calculate side ratios to establish similarity.

The ratio of AD to AC is 8/16 = 1/2, and the ratio of AE to AB is 10/20 = 1/2.

Checking if the corresponding sides surrounding the shared angle are in the same proportion.

2

Apply the Side-Angle-Side (SAS) similarity theorem.

Triangle ADE is similar to triangle ACB (triangle ADE ~ triangle ACB), where vertex A corresponds to A, D corresponds to C, and E corresponds to B.

Since the ratio of two pairs of corresponding sides is equal and their included angle is congruent, the triangles are similar.

3

Determine the area ratio based on the similarity scale factor.

The ratio of the area of triangle ADE to the area of triangle ACB is (1/2)^2 = 1/4.

The ratio of the areas of two similar figures is equal to the square of their similarity ratio.

4

Relate the area of the quadrilateral to the area of the smaller triangle.

Area(BCED) = Area(ACB) - Area(ADE) = 4 * Area(ADE) - Area(ADE) = 3 * Area(ADE).

The area of the quadrilateral is the difference between the areas of the larger and smaller triangles.

5

Solve for the area of triangle ADE.

Area(ADE) = 54 / 3 = 18.

Dividing the given area of the quadrilateral by 3 yields the area of the smaller triangle.

Anahtar Kavram

SAS Triangle Similarity and the Area Ratios of Similar Triangles

Tahmini Süre:2m 30s

Soru 563Soru →

An educational organization wants to estimate the number of high school seniors in a school district who plan to major in a STEM field. A random sample of $320$ high school seniors in the district was surveyed, and $35\%$ of the surveyed students reported that they plan to major in a STEM field. The survey has an associated margin of error of $4\%$ . There are $4,500$ high school seniors in the school district. Based on these results, what is the maximum estimated number of high school seniors in the district who plan to major in a STEM field?

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Cevap: 1755

Cevap

1755

The survey estimate for the proportion of students who plan to major in a STEM field is

35\%

with a margin of error of

4\%

. This means the true proportion is estimated to be between

35\% - 4\% = 31\%

and

35\% + 4\% = 39\%

. To find the maximum estimated number of students in the population of

4,500

, we use the maximum estimated proportion of

39\%

(or

0.39

). Multiplying

0.39

by

4,500

yields

1,755

.

Adım Adım Çözüm

1

Calculate the maximum estimated percentage of students by adding the margin of error to the sample percentage.

39\%

(or

0.39

)

The margin of error gives the range of values above and below the sample estimate that represents the true population proportion. The maximum value of this range is found by adding the margin of error.

2

Multiply the maximum estimated proportion by the total number of high school seniors in the district.

1,755

To generalize the sample results to the entire population of

4,500

seniors, multiply the maximum proportion by the total population size.

Anahtar Kavram

Estimating population parameters using sample proportions and margin of error

Soru 564Soru →

In the $xy$ -plane, an angle $\theta$ is in standard position. The terminal ray of $\theta$ is rotated counterclockwise by $\frac{7\pi}{6}$ radians, and then rotated clockwise by $135^\circ$ . If the terminal ray of the resulting angle lies on the line $y = -x$ in the fourth quadrant, and the original angle $\theta$ has a measure of $d$ degrees, where $0 \leq d < 360$ , what is the value of $d$ ?

Cevabı ve açıklamayı göster

Cevap: 240

Cevap

The value of

d

is 240.

A terminal ray in the fourth quadrant lying on the line

y = -x

forms a

315^\circ

angle in standard position. The first rotation is counterclockwise by

\frac{7\pi}{6}

radians, which is equivalent to

\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ

. The second rotation is clockwise by

135^\circ

, representing a decrease of

135^\circ

. Therefore, the total transformation is

\theta + 210^\circ - 135^\circ = 315^\circ

. Solving for

\theta

gives

\theta + 75^\circ = 315^\circ

, which simplifies to

\theta = 240^\circ

. Since

240^\circ

lies in the interval

[0, 360)

, the value of

d

is 240.

Adım Adım Çözüm

1

Determine the angle in standard position for a terminal ray on the line

y = -x

in the fourth quadrant.

The terminal ray corresponds to an angle of

315^\circ

(or any angle coterminal with it).

The line

y = -x

in the fourth quadrant makes an angle of

45^\circ

below the positive

x

-axis, which corresponds to

360^\circ - 45^\circ = 315^\circ

in standard position.

2

Convert the counterclockwise rotation from radians to degrees.

\frac{7\pi}{6} \text{ radians} = 210^\circ

.

To convert radians to degrees, multiply the angle in radians by

\frac{180^\circ}{\pi}

.

3

Express the rotations mathematically and set up the equation for

\theta

.

\theta + 210^\circ - 135^\circ = 315^\circ + 360^\circ n

(where

n

is an integer).

In standard position, counterclockwise rotations represent positive changes in angle measure, whereas clockwise rotations represent negative changes in angle measure.

4

Solve for

\theta

and apply the domain restriction

0 \leq d < 360

.

\theta = 240^\circ

, so

d = 240

.

Simplifying the equation gives

\theta + 75^\circ = 315^\circ

, which yields

\theta = 240^\circ

when

n=0

.

Anahtar Kavram

Converting angles from radians to degrees, understanding the direction of rotation, and determining standard position angles on the coordinate plane.

Soru 565Soru →

In the $xy$ -plane, the graph of the polynomial function $g$ , defined by $g(x) = 3(x - k)(x + 2)^2$ , has a $y$ -intercept at $(0, -24)$ , where $k$ is a constant. What is the value of $k$ ?

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Cevap: 2

Cevap

The correct answer is 2.

Evaluating the function at

x = 0

yields

g(0) = 3(-k)(2)^2 = -12k

. Setting this equal to the given

y

-intercept value of

-24

gives

-12k = -24

. Solving for

k

gives

k = 2

.

Adım Adım Çözüm

1

Determine the value of the function at the

y

-intercept

g(0) = -24

The

y

-intercept of a graph in the

xy

-plane is the point where

x = 0

. Since the

y

-intercept is

(0, -24)

, the function value when

x = 0

must be

-24

.

2

Substitute

x = 0

into the function definition

g(0) = 3(0 - k)(0 + 2)^2

To evaluate the expression at

x = 0

, we substitute

0

for every instance of

x

in the equation.

3

Simplify the algebraic expression

g(0) = -12k

Simplifying the terms:

(0+2)^2 = 4

, and

3(0-k) = -3k

. Multiplying these gives

-3k \times 4 = -12k

.

4

Solve for the constant

k

k = 2

Equating the simplified expression to the known

y

-value at the intercept gives

-12k = -24

. Dividing both sides by

-12

yields

k = 2

.

Anahtar Kavram

Using the y-intercept of a polynomial function to solve for an unknown constant coefficient.

Soru 566Soru →

A network router transmits data from two sources, Server $A$ and Server $B$ , at a constant ratio of $5$ megabytes ( $\text{MB}$ ) from Server $A$ for every $3\text{ MB}$ from Server $B$ . In a single session, the router transmitted a combined total of $160\text{ MB}$ of data from both servers. During the next session, the router transmits data at the same ratio, but the amount of data transmitted from Server $A$ is $20\%$ greater than the amount of data transmitted from Server $A$ in the first session. How many megabytes of data are transmitted from Server $B$ in the second session?

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Cevap: 72

Cevap

The amount of data transmitted from Server B in the second session is 72 megabytes.

To find the data transmitted from Server B in the second session, we first determine the individual data amounts from the first session. Given the ratio of data from Server A to Server B is

5:3

and the total is

160\text{ MB}

, we can express their data amounts as

5x

and

3x

. Solving

5x + 3x = 160

yields

x = 20

. Thus, Server A transmitted

5 \times 20 = 100\text{ MB}

and Server B transmitted

3 \times 20 = 60\text{ MB}

in the first session. In the second session, the data from Server A increased by

20\%

, yielding

100 \times 1.20 = 120\text{ MB}

. Since the transmission ratio between Server A and Server B remains constant at

5:3

, we can set up the proportion

\frac{120}{B} = \frac{5}{3}

. Cross-multiplying gives

5B = 360

, which solves to

B = 72

megabytes.

Adım Adım Çözüm

1

Set up an equation for the total data in the first session.

5x + 3x = 160 \implies 8x = 160 \implies x = 20

The ratio of data from Server A to Server B is

5:3

, meaning the amounts can be represented as

5x

and

3x

for some multiplier

x

.

2

Calculate the data transmitted from Server A in the first session.

A_1 = 5(20) = 100\text{ MB}

Multiply the ratio term for Server A by the scale factor

x

to find its initial data contribution.

3

Calculate the increased data from Server A in the second session.

A_2 = 100 \times (1 + 0.20) = 120\text{ MB}

Increase the first session's data from Server A by

20\%

.

4

Use the constant ratio to find the data from Server B in the second session.

B_2 = 72\text{ MB}

Set up the proportion

\frac{120}{B_2} = \frac{5}{3}

and solve for

B_2

by cross-multiplying.

Anahtar Kavram

Solving part-to-part and part-to-whole ratio problems under proportional changes

Soru 567Soru →

In the $xy$ -plane, the graph of $y = x^2 - 8x + 12$ represents the quadratic function $f$ . If this graph is shifted $3$ units to the left and $5$ units up to create the graph of a new function $g$ , what is the $y$ -value of the vertex of the graph of $g$ ?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The y-value of the vertex of the graph of g is 1.

The vertex of the original quadratic function f(x) = x^2 - 8x + 12 can be found by rewriting it in vertex form, which is f(x) = (x - 4)^2 - 4. This shows that the vertex of the original graph is (4, -4). A translation of 3 units to the left subtracts 3 from the x-coordinate of the vertex (4 - 3 = 1), and a translation of 5 units up adds 5 to the y-coordinate of the vertex (-4 + 5 = 1). Thus, the vertex of the graph of the new function g is (1, 1), making its y-value 1.

Adım Adım Çözüm

1

Find the vertex of the original quadratic function.

The vertex of the original graph is

(4, -4)

.

By completing the square on

y = x^2 - 8x + 12

, we get

y = (x - 4)^2 - 4

, which reveals the vertex is at

(4, -4)

.

2

Apply the translation to the vertex coordinates.

The translated vertex coordinates are

(1, 1)

.

Shifting a point

(x, y)

by

3

units left and

5

units up results in the point

(x - 3, y + 5)

. Applying this to the vertex

(4, -4)

yields

(4 - 3, -4 + 5) = (1, 1)

.

3

Identify the y-coordinate of the new vertex.

The y-value is

1

.

The vertex of the graph of

g

is

(1, 1)

, where the second coordinate represents the y-value.

Anahtar Kavram

Identifying the vertex of a quadratic function and applying horizontal and vertical translations in the coordinate plane.

Soru 568Soru →

In the $xy$ -plane, the graph of the equation $x^2 - 8x + y^2 = 0$ is a circle. What is the radius of the circle?

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Cevap: 4

Cevap

The radius of the circle is 4.

By completing the square on the

x

-terms in the equation

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

, we add

( -8/2 )^2 = 16

to both sides, yielding

(x - 4)^2 + y^2 = 16

. Comparing this to the standard equation of a circle,

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

, we find that

r^2 = 16

. Taking the square root of both sides gives a radius of

4

.

Adım Adım Çözüm

1

Group the

x

-terms together and prepare to complete the square.

(x^2 - 8x) + y^2 = 0

To write the equation in the standard form of a circle,

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

, we need to complete the square for the quadratic expression in

x

.

2

Complete the square for the

x

terms by adding

16

to both sides of the equation.

(x^2 - 8x + 16) + y^2 = 16

, which simplifies to

(x - 4)^2 + y^2 = 16

.

Adding

( -8/2 )^2 = 16

to both sides creates a perfect square trinomial

(x-4)^2

on the left side of the equation.

3

Identify the value of

r^2

and find the radius

r

.

r^2 = 16

, which gives

r = \sqrt{16} = 4

.

Comparing the equation

(x - 4)^2 + y^2 = 16

to the standard form

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

shows that

r^2 = 16

. Since the radius must be positive,

r = 4

.

Anahtar Kavram

Completing the square to find the standard form equation of a circle,

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

, to determine its radius.

Soru 569Soru →

In right triangle $XYZ$ , the measure of angle $Y$ is $90^\circ$ . If $\cos(X) + \sin(Z) = 1.6$ , what is the value of $\sin(X)$ ?

Cevabı ve açıklamayı göster

Cevap: 0.6

Cevap

0.6

The two acute angles in a right triangle sum to

90^\circ

, which means they are complementary. Therefore, the co-function identity

\sin(Z) = \cos(X)

applies. Substituting this relationship into the equation

\cos(X) + \sin(Z) = 1.6

gives

2\cos(X) = 1.6

, leading to

\cos(X) = 0.8

. Finally, using the Pythagorean identity

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

, we find

\sin(X) = \sqrt{1 - 0.8^2} = 0.6

.

Adım Adım Çözüm

1

Determine the relationship between angles

X

and

Z

.

X + Z = 90^\circ

Since the sum of the angles in right triangle

XYZ

is

180^\circ

and angle

Y

is

90^\circ

, the two acute angles

X

and

Z

must sum to

90^\circ

, making them complementary.

2

Use the co-function identity to relate

\sin(Z)

and

\cos(X)

.

\sin(Z) = \cos(X)

For complementary angles, the sine of one angle is equal to the cosine of the other.

3

Substitute the identity into the given equation to solve for

\cos(X)

.

\cos(X) = 0.8

Replacing

\sin(Z)

with

\cos(X)

in

\cos(X) + \sin(Z) = 1.6

yields

2\cos(X) = 1.6

, so

\cos(X) = 0.8

.

4

Solve for

\sin(X)

using the Pythagorean identity.

\sin(X) = 0.6

Since

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

, we have

\sin^2(X) + (0.8)^2 = 1

, which gives

\sin^2(X) = 1 - 0.64 = 0.36

. Since

X

is an acute angle, its sine must be positive, so

\sin(X) = 0.6

.

Anahtar Kavram

Co-function and Pythagorean trigonometric identities in right triangles

Soru 570Soru →

A metallic plate is in the shape of a trapezoid. The parallel sides of the plate have lengths of $9$ centimeters and $30$ centimeters. The two non-parallel sides have lengths of $10$ centimeters and $17$ centimeters. What is the area, in square centimeters, of the metallic plate?

Cevabı ve açıklamayı göster

Cevap: 156

Cevap

156

To find the area of the trapezoid, we can find its height by drawing perpendicular lines from the vertices of the shorter base to the longer base. This splits the trapezoid into a rectangle of width

9\text{ cm}

and two right triangles with hypotenuses of

10\text{ cm}

and

17\text{ cm}

. The sum of the bases of these two triangles is

30 - 9 = 21\text{ cm}

. Letting their bases be

x

and

y

(where

x + y = 21

), we apply the Pythagorean theorem:

h^2 + x^2 = 100

and

h^2 + y^2 = 289

. Subtracting the equations yields

y^2 - x^2 = 189

. Since

y^2 - x^2 = (y - x)(y + x)

, we have

21(y - x) = 189

, which simplifies to

y - x = 9

. Solving the system

x + y = 21

and

y - x = 9

yields

x = 6

and

y = 15

. Using the Pythagorean triple

6

-

8

-

10

, the height

h

is

8\text{ cm}

. The area is

\frac{1}{2}(9 + 30)(8) = 156

square centimeters.

Adım Adım Çözüm

1

Drop perpendicular lines (heights) from the endpoints of the shorter base to the longer base.

The trapezoid is decomposed into a central rectangle with a width of

9

centimeters and two right triangles with bases

x

and

y

and hypotenuses of

10

centimeters and

17

centimeters, respectively.

This sets up a system of equations relating the heights and bases of the right triangles.

2

Determine the relationship between the bases of the two right triangles.

The sum of the bases of the two right triangles is

x + y = 30 - 9 = 21

centimeters.

The sum of the bases of the right triangles plus the width of the rectangle equals the total length of the longer base of the trapezoid.

3

Set up equations using the Pythagorean theorem for the two right triangles.

h^2 + x^2 = 100

and

h^2 + y^2 = 289

.

Both right triangles share the same height

h

of the trapezoid.

4

Solve for the difference between the two triangle bases.

Subtracting the first equation from the second gives

y^2 - x^2 = 189

. Factoring yields

(y - x)(y + x) = 189

. Substituting

y + x = 21

yields

21(y - x) = 189 \implies y - x = 9

.

Subtracting the equations eliminates the height variable, allowing us to find the difference between the bases.

5

Solve the system of linear equations for

x

and

y

.

Adding

x + y = 21

and

y - x = 9

gives

2y = 30 \implies y = 15

. Substituting back gives

x = 6

.

This determines the exact base segments of both right triangles.

6

Calculate the height of the trapezoid.

h = \sqrt{10^2 - 6^2} = \sqrt{64} = 8

centimeters.

The height of the trapezoid is required to calculate its area.

7

Compute the area of the trapezoid.

\text{Area} = \frac{1}{2}(9 + 30)(8) = 156

square centimeters.

This uses the standard formula for the area of a trapezoid.

Anahtar Kavram

Decomposing a non-isosceles trapezoid into a rectangle and two right triangles to solve for the height using systems of quadratic equations derived from the Pythagorean theorem.

Tahmini Süre:1m 30s

Soru 571Soru →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . Point $D$ lies on side $AC$ and point $E$ lies on hypotenuse $AB$ such that segment $DE$ is perpendicular to $AB$ . The length of segment $AE$ is $x$ , the length of segment $AD$ is $x + 1$ , the length of segment $CD$ is $2$ , and the length of segment $BE$ is $5$ . What is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

By identifying that triangle

AED

is similar to triangle

ACB

(due to shared angle

A

and right angles at

E

and

C

), we can set up the proportion

\frac{AE}{AC} = \frac{AD}{AB}

. Expressing the segments as

AC = x + 3

and

AB = x + 5

leads to the equation

\frac{x}{x + 3} = \frac{x + 1}{x + 5}

. Solving this equation yields

x = 3

.

Adım Adım Çözüm

1

Establish similarity between triangles

AED

and

ACB

.

\triangle AED \sim \triangle ACB

Both triangles share the acute angle

A

(

\angle EAD = \angle CAB

) and both have a right angle (

\angle AED = \angle ACB = 90^\circ

), satisfying the Angle-Angle (AA) similarity criterion.

2

Write the proportion of corresponding sides.

\frac{AE}{AC} = \frac{AD}{AB}

In similar triangles, the ratios of the lengths of corresponding sides are equal.

3

Express the total side lengths of triangle

ABC

using segment addition.

AC = x + 3

and

AB = x + 5

Since

D

is on

AC

,

AC = AD + CD = (x + 1) + 2 = x + 3

. Since

E

is on

AB

,

AB = AE + BE = x + 5

.

4

Substitute the algebraic expressions into the side ratio proportion.

\frac{x}{x + 3} = \frac{x + 1}{x + 5}

Substituting

AE = x

,

AD = x + 1

,

AC = x + 3

, and

AB = x + 5

into the similarity proportion.

5

Solve the proportion for

x

.

x = 3

Cross-multiplying gives

x(x + 5) = (x + 1)(x + 3) \implies x^2 + 5x = x^2 + 4x + 3

. Subtracting

x^2

and

4x

from both sides results in

x = 3

.

Anahtar Kavram

Identifying similar right triangles via the AA similarity criterion and solving resulting algebraic proportions.

Soru 572Soru →

At a currency exchange booth, $4$ US dollars (USD) can be exchanged for $5$ local credits, and $6$ local credits can be exchanged for $15$ reward coupons. Based on these rates, what is the total number of reward coupons that can be obtained in exchange for $24$ USD?

Cevabı ve açıklamayı göster

Cevap: 75

Cevap

75

To find the number of reward coupons obtained from

24

USD, we perform a two-step rate conversion. First, convert USD to local credits:

24\text{ USD} \times \frac{5\text{ local credits}}{4\text{ USD}} = 30\text{ local credits}

. Second, convert local credits to reward coupons:

30\text{ local credits} \times \frac{15\text{ coupons}}{6\text{ local credits}} = 75\text{ coupons}

. Thus, the final value is

75

.

Adım Adım Çözüm

1

Set up a proportion to find the number of local credits equivalent to

24

USD.

30

local credits

Using the exchange rate of

4

USD to

5

local credits, multiply the starting value of

24

USD by the conversion factor

\frac{5}{4}

.

2

Set up a proportion to find the number of coupons equivalent to

30

local credits.

75

coupons

Using the exchange rate of

6

local credits to

15

coupons, multiply the

30

local credits by the conversion factor

\frac{15}{6}

.

Anahtar Kavram

Multi-step rate conversion and unit analysis

Soru 573Soru →

Two connected gears, Gear A and Gear B, rotate together such that the belt connecting them does not slip. The radius of Gear A is $15$ centimeters and the radius of Gear B is $9$ centimeters. If Gear A rotates through a central angle of $\frac{4\pi}{15}$ radians, Gear B rotates through a central angle of $x$ degrees. What is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

80

The arc length

s

that a point on the belt travels is given by the product of the radius and the angle in radians of Gear A:

s = 15 \times \frac{4\pi}{15} = 4\pi

cm. Since the belt does not slip, Gear B rotates through the same arc length. The angle of Gear B in radians is

\theta = \frac{4\pi}{9}

radians. To convert this angle to degrees, multiply by

\frac{180}{\pi}

to get

\frac{4\pi}{9} \times \frac{180}{\pi} = 80

degrees.

Adım Adım Çözüm

1

Calculate the arc length of the rotation for Gear A using the formula

s = r\theta

.

s = 4\pi

centimeters

To find the distance a point on the belt travels, which is shared by both gears.

2

Determine the rotation angle of Gear B in radians using the arc length and Gear B's radius.

\theta_B = \frac{4\pi}{9}

radians

Because the belt does not slip, Gear B must rotate by the same linear arc length as Gear A.

3

Convert the angle of Gear B from radians to degrees by multiplying by

\frac{180}{\pi}

.

x = 80

To find the measure of the angle in degrees as requested by the question.

Anahtar Kavram

Converting central angles between radians and degrees in the context of arc lengths of connected circles.

Soru 574Soru →

The table below shows selected values of $x$ and the corresponding values of the polynomial function $p(x)$ .

$x$	$p(x)$
$0$	$-16$
$2$	$0$
$3$	$-4$
$4$	$0$

If $p(x) = a(x-2)(x-4)(x-k)$ for all real numbers $x$ , where $a$ and $k$ are constants, what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The value of

k

is

1

.

Substituting the coordinate pair

(0, -16)

into the equation

p(x) = a(x-2)(x-4)(x-k)

gives

p(0) = a(-2)(-4)(-k) = -8ak = -16

, which simplifies to

ak = 2

. Substituting the coordinate pair

(3, -4)

gives

p(3) = a(3-2)(3-4)(3-k) = -a(3-k) = -4

. Expanding this equation yields

-3a + ak = -4

. Replacing

ak

with

2

in this equation results in

-3a + 2 = -4

, which simplifies to

-3a = -6

, so

a = 2

. Using

a = 2

in the relation

ak = 2

gives

2k = 2

, which yields

k = 1

.

Adım Adım Çözüm

1

Substitute the coordinates

(0, -16)

from the table into the given equation

p(x) = a(x-2)(x-4)(x-k)

to form a relationship between the constants.

p(0) = a(0-2)(0-4)(0-k) = a(-2)(-4)(-k) = -8ak = -16

, which simplifies to

ak = 2

.

To establish a relation between constants

a

and

k

using the y-intercept of the polynomial.

2

Substitute another known point from the table,

(3, -4)

, into the polynomial equation.

p(3) = a(3-2)(3-4)(3-k) = a(1)(-1)(3-k) = -a(3-k) = -4

.

To obtain a second equation relating the constants

a

and

k

.

3

Solve the system of equations by distributing

-a

in the second equation and substituting the value of

ak

from the first step.

-3a + ak = -4

. Since

ak = 2

, this becomes

-3a + 2 = -4 \implies -3a = -6 \implies a = 2

.

To solve for the leading coefficient constant

a

.

4

Substitute the value of

a

back into the first relation to solve for

k

.

Since

a = 2

and

ak = 2

, we have

2k = 2 \implies k = 1

.

To find the final value of

k

.

Anahtar Kavram

Solving for polynomial constants using given coordinate points from a table

Soru 575Soru →

A cubic polynomial function $f$ has $x$ -intercepts at $(2, 0)$ , $(5, 0)$ , and $(c, 0)$ , where $c$ is a positive constant. In the $xy$ -plane, the graph of $y = f(x)$ has a $y$ -intercept at $(0, 60)$ . If $f(1) = 16$ , what is the value of $c$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The correct answer is 3.

The factored form of a cubic polynomial with

x

-intercepts at

x = 2

,

x = 5

, and

x = c

is

f(x) = a(x-2)(x-5)(x-c)

. Evaluating the function at the

y

-intercept

x = 0

gives

f(0) = a(-2)(-5)(-c) = -10ac = 60

, which simplifies to

ac = -6

. Using the point

(1, 16)

yields

f(1) = a(1-2)(1-5)(1-c) = 4a(1-c) = 16

. Expanding this equation gives

4a - 4ac = 16

. Substituting

ac = -6

into this equation yields

4a - 4(-6) = 16

, which simplifies to

4a + 24 = 16

, resulting in

a = -2

. Finally, substituting

a = -2

into

ac = -6

yields

-2c = -6

, so

c = 3

.

Adım Adım Çözüm

1

Write the polynomial in factored form using its

x

-intercepts.

f(x) = a(x-2)(x-5)(x-c)

By the factor theorem, if a polynomial has

x

-intercepts at

x = r_1, r_2, r_3

, then

(x-r_1)

,

(x-r_2)

, and

(x-r_3)

are factors of the polynomial.

2

Use the

y

-intercept

(0, 60)

to find a relation between

a

and

c

.

ac = -6

Since the

y

-intercept is at

(0, 60)

, we substitute

x = 0

into the polynomial and set the expression equal to

60

:

a(-2)(-5)(-c) = -10ac = 60

, which gives

ac = -6

.

3

Use the given point

f(1) = 16

to set up a second equation.

4a(1-c) = 16

Substitute

x = 1

and

f(1) = 16

into the factored form:

a(1-2)(1-5)(1-c) = a(-1)(-4)(1-c) = 4a(1-c) = 16

.

4

Solve for the leading coefficient

a

by substituting

ac = -6

.

a = -2

Expanding the equation from Step 3 yields

4a - 4ac = 16

. Substituting

ac = -6

gives

4a - 4(-6) = 16

, which simplifies to

4a + 24 = 16

, so

4a = -8

and

a = -2

.

5

Solve for the constant

c

.

c = 3

Using the relation

ac = -6

and substituting

a = -2

gives

-2c = -6

, which yields

c = 3

.

Anahtar Kavram

Using the factor theorem to set up a cubic polynomial equation and solving for unknown parameters using given points.

Soru 576Soru →

The measure of angle $A$ is $45^\circ$ greater than the measure of angle $B$ . If the measure of angle $B$ is $\frac{5\pi}{12}$ radians, what is the measure of angle $A$ , in degrees?

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

120

The correct answer is 120. First, convert the measure of angle

B

from radians to degrees:

\frac{5\pi}{12} \times \frac{180}{\pi} = \frac{5 \times 180}{12} = 5 \times 15 = 75^\circ

. Since angle

A

is

45^\circ

greater than angle

B

, add

45^\circ

to the measure of angle

B

:

75^\circ + 45^\circ = 120^\circ

.

Adım Adım Çözüm

1

Convert the measure of angle

B

from radians to degrees.

Angle

B

has a measure of

75^\circ

.

To convert from radians to degrees, multiply the radian measure by

\frac{180}{\pi}

.

2

Calculate the measure of angle

A

by adding

45^\circ

to the measure of angle

B

.

Angle

A

has a measure of

120^\circ

.

It is given that the measure of angle

A

is

45^\circ

greater than the measure of angle

B

.

Anahtar Kavram

To convert an angle from radians to degrees, multiply the angle measure in radians by

\frac{180}{\pi}

.

Tahmini Süre:1m 0s

Soru 577Soru →

A solar energy storage system collects electricity from two solar panel arrays, Array $X$ and Array $Y$ . Array $X$ produces $3$ kilowatt-hours (kWh) of electricity for every $4$ hours of direct sunlight. Array $Y$ produces $5$ kWh of electricity for every $6$ hours of direct sunlight. If both arrays receive direct sunlight simultaneously, how many hours of direct sunlight are required for the two arrays to produce a combined total of $38$ kWh of electricity?

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

24

The correct answer is

24

. First, find the rate of each array in kWh per hour: Array

X

has a rate of

\frac{3}{4}

kWh/hour and Array

Y

has a rate of

\frac{5}{6}

kWh/hour. Since the arrays operate simultaneously, add their individual rates to find the combined rate of

\frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12}

kWh/hour. To find the total hours

t

needed to produce

38

kWh, set up the equation

\frac{19}{12}t = 38

. Solving for

t

gives

t = 38 \times \frac{12}{19} = 24

hours.

Adım Adım Çözüm

1

Calculate the individual hourly rates of electricity production for Array

X

and Array

Y

.

Array

X

produces at a rate of

\frac{3}{4}

kWh per hour, and Array

Y

produces at a rate of

\frac{5}{6}

kWh per hour.

Converting the given production rates to unit rates (kWh per hour) allows them to be directly compared and combined.

2

Add the individual rates to find the combined rate of both arrays operating simultaneously.

Combined rate is

\frac{19}{12}

kWh per hour.

Since both arrays operate simultaneously, their rates of production add together. The common denominator for

4

and

6

is

12

, so

\frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12}

.

3

Determine the time required to produce a combined total of

38

kWh.

24

hours.

Divide the total target production of

38

kWh by the combined rate of

\frac{19}{12}

kWh per hour:

38 \div \frac{19}{12} = 38 \times \frac{12}{19} = 2 \times 12 = 24

.

Anahtar Kavram

Ratios, Rates, and Proportions

Tahmini Süre:1m 30s

Soru 578Soru →

A surveyor maps a triangular plot of land, $LMN$ . A boundary line segment is drawn from point $P$ on side $LN$ to point $Q$ on side $LM$ , creating a smaller triangular section $LPQ$ . The measure of angle $LMN$ is equal to the measure of angle $LPQ$ . The surveyed lengths are $LM = 18$ meters, $LP = 8$ meters, and $LQ = 12$ meters. What is the length, in meters, of segment $PN$ ?

Cevabı ve açıklamayı göster

Cevap: 19

Cevap

19

By identifying that triangle

LMN

and triangle

LPQ

share the angle at vertex

L

and have another pair of congruent angles (

\angle LMN = \angle LPQ

), we establish similarity between the two triangles:

\triangle LMN \sim \triangle LPQ

. Using the proportional relationship of their corresponding sides, we write

\frac{LM}{LP} = \frac{LN}{LQ}

. Substituting the given values

LM = 18

,

LP = 8

, and

LQ = 12

yields

\frac{18}{8} = \frac{LN}{12}

, which solves to

LN = 27

. Finally, subtracting

LP = 8

from the total length of segment

LN

gives the length of segment

PN

as

19

meters.

Adım Adım Çözüm

1

Identify similar triangles.

Triangle

LMN

is similar to triangle

LPQ

(

\triangle LMN \sim \triangle LPQ

).

They share angle

L

(

\angle MLN = \angle PLQ

) and we are given that

\angle LMN = \angle LPQ

. By the Angle-Angle (AA) similarity criterion, the two triangles are similar.

2

Set up the ratio of corresponding sides.

\frac{LM}{LP} = \frac{LN}{LQ}

Corresponding sides of similar triangles are proportional.

3

Solve for the length of side

LN

.

LN = 27

meters

Substituting

LM = 18

,

LP = 8

, and

LQ = 12

into the proportion yields

\frac{18}{8} = \frac{LN}{12}

. Solving for

LN

gives

LN = 12 \times \frac{18}{8} = 27

.

4

Calculate the length of segment

PN

.

PN = 19

meters

Point

P

lies on segment

LN

, so the length of segment

PN

is the difference between

LN

and

LP

, which is

27 - 8 = 19

.

Anahtar Kavram

Triangle similarity using the Angle-Angle (AA) criterion and proportional side ratios.

Soru 579Soru →

A pendulum swings through an angle of $40^\circ$ , and the tip of the pendulum travels an arc of length $8\pi$ inches. What is the length of the pendulum, in inches?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

36

To find the length of the pendulum, which represents the radius

r

of the circular path it sweeps, we can use the arc length formula

s = r\theta

, where

s

is the arc length and

\theta

is the central angle in radians. First, convert the given angle from degrees to radians:

\theta = 40^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{9}

radians. Next, substitute the arc length

s = 8\pi

and the angle

\theta = \frac{2\pi}{9}

into the formula:

8\pi = r \left(\frac{2\pi}{9}\right)

. Solving for

r

by multiplying both sides by

\frac{9}{2\pi}

gives

r = 36

. Alternatively, you can use the ratio of the sector's central angle to the total angle of a circle:

\frac{40^\circ}{360^\circ} = \frac{1}{9}

. This means the arc length is

\frac{1}{9}

of the circumference of the circle:

8\pi = \frac{1}{9}(2\pi r)

. Dividing both sides by

2\pi

yields

4 = \frac{1}{9}r

, so

r = 36

.

Adım Adım Çözüm

1

Convert the swing angle of the pendulum from degrees to radians.

\theta = \frac{2\pi}{9}

radians

The arc length formula

s = r\theta

requires the angle

\theta

to be in radians.

2

Set up the arc length equation using

s = r\theta

, where

s = 8\pi

is the arc length and

r

is the length of the pendulum.

8\pi = r \left(\frac{2\pi}{9}\right)

The tip of the pendulum travels along a circular path whose radius is the length of the pendulum.

3

Solve the equation for the radius

r

.

r = 36

Isolating

r

by multiplying both sides by

\frac{9}{2\pi}

yields the length of the pendulum.

Anahtar Kavram

Converting angle measures between degrees and radians and applying the arc length formula.

Soru 580Soru →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . A point $D$ lies on side $AC$ such that $BD$ is the angle bisector of angle $ABC$ . If $BC = 28$ and $BD = 35$ , what is the length of segment $AD$ ?

Cevabı ve açıklamayı göster

Cevap: 75

Cevap

The length of segment

AD

is 75.

By applying the Pythagorean theorem to right triangle

BCD

, we find

CD = 21

. Then, by the angle bisector theorem, the ratio of

AB

to

BC

equals the ratio of

AD

to

CD

, which gives

AB = \frac{4}{3}AD

. Using the Pythagorean theorem on right triangle

ABC

, we solve

(\frac{4}{3}AD)^2 = 28^2 + (21 + AD)^2

to find the positive length

AD = 75

.

Adım Adım Çözüm

1

Find the length of

CD

using the Pythagorean theorem on right triangle

BCD

.

CD = 21

Since angle

C

is a right angle, triangle

BCD

is a right triangle with hypotenuse

BD

and leg

BC

.

2

Apply the angle bisector theorem to express

AB

in terms of

AD

.

AB = \frac{4}{3}AD

The angle bisector theorem states that

\frac{AD}{CD} = \frac{AB}{BC}

. Substituting

BC = 28

and

CD = 21

gives

\frac{AD}{21} = \frac{AB}{28}

.

3

Set up a quadratic equation using the Pythagorean theorem on right triangle

ABC

.

(\frac{4}{3}AD)^2 = 28^2 + (21 + AD)^2

In right triangle

ABC

, the hypotenuse is

AB

and the legs are

BC = 28

and

AC = CD + AD = 21 + AD

.

4

Solve the quadratic equation for

AD

.

AD = 75

Expanding and simplifying the equation yields

AD^2 - 54AD - 1575 = 0

, which factors as

(AD - 75)(AD + 21) = 0

. Since length must be positive,

AD = 75

.

Anahtar Kavram

Pythagorean Theorem and Angle Bisector Theorem

Alternatif Yöntem

Let

\theta = \angle DBC

. Since

BD

bisects angle

B

,

\angle ABC = 2\theta

. In right triangle

BCD

,

\cos\theta = \frac{BC}{BD} = \frac{28}{35} = \frac{4}{5}

. In right triangle

ABC

,

\cos(2\theta) = \frac{BC}{AB} = \frac{28}{AB}

. Using the double-angle identity

\cos(2\theta) = 2\cos^2\theta - 1

, we get

\frac{28}{AB} = 2(\frac{4}{5})^2 - 1 = \frac{7}{25}

, which gives

AB = 100

. Finally,

AC = \sqrt{AB^2 - BC^2} = \sqrt{100^2 - 28^2} = 96

, so

AD = AC - CD = 96 - 21 = 75

.

Tahmini Süre:2m 30s

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