Geometry and Trigonometry Alıştırma Soruları — SAT — Sayfa 2

A sphere has a surface area of $144\pi$ square inches. What is the volume, in cubic inches, of the sphere?

48\pi

216\pi

288\pi

864\pi

Cevabı ve açıklamayı göster

Cevap: $288\pi$

Cevap

The volume of the sphere is

288\pi

cubic inches.

The surface area of a sphere is given by

A = 4\pi r^2

, where

r

is the radius. Setting this equal to the given surface area yields

4\pi r^2 = 144\pi

, which simplifies to

r^2 = 36

, meaning the radius of the sphere is

6

inches. The volume of a sphere is given by

V = \frac{4}{3}\pi r^3

. Substituting

r = 6

into this formula gives

V = \frac{4}{3}\pi (6^3) = \frac{4}{3}\pi (216) = 288\pi

cubic inches.

Adım Adım Çözüm

1

Set the formula for the surface area of a sphere,

A = 4\pi r^2

, equal to the given surface area of

144\pi

.

4\pi r^2 = 144\pi

, which simplifies to

r^2 = 36

.

To find the square of the radius of the sphere.

2

Take the square root of both sides to find the radius

r

.

r = 6

.

The radius of a sphere must be a positive number.

3

Substitute the radius

r = 6

into the formula for the volume of a sphere,

V = \frac{4}{3}\pi r^3

.

V = \frac{4}{3}\pi(6^3) = \frac{4}{3}\pi(216) = 288\pi

.

To compute the final volume of the sphere.

Anahtar Kavram

The volume of a sphere can be calculated by first finding its radius from its surface area using the formulas

A = 4\pi r^2

and

V = \frac{4}{3}\pi r^3

.

Soru 22Soru →

A solid sphere has a volume of $36\pi$ cubic centimeters. What is the radius, in centimeters, of the sphere?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The radius of the sphere is 3 centimeters.

The volume of a sphere is given by the formula

V = \frac{4}{3}\pi r^3

. Substituting

36\pi

for

V

gives

36\pi = \frac{4}{3}\pi r^3

. Dividing both sides by

\pi

results in

36 = \frac{4}{3}r^3

. Multiplying both sides by

\frac{3}{4}

yields

27 = r^3

. Taking the cube root of both sides gives

r = 3

. Thus, the radius of the sphere is 3 centimeters.

Adım Adım Çözüm

1

State the sphere volume formula.

V = \frac{4}{3}\pi r^3

The volume of a sphere is calculated using this standard formula relating volume to radius.

2

Substitute the given volume value into the formula.

36\pi = \frac{4}{3}\pi r^3

The problem specifies the volume is

36\pi

cubic centimeters.

3

Solve for the radius cubed (

r^3

).

r^3 = 27

Dividing both sides of the equation by

\pi

yields

36 = \frac{4}{3}r^3

, and multiplying both sides by

\frac{3}{4}

isolates

r^3

to give 27.

4

Take the cube root of both sides to find the radius (

r

).

r = 3

The cube root of 27 is 3, since

3 \times 3 \times 3 = 27

.

Anahtar Kavram

Using the volume of a sphere formula to solve for an unknown radius.

Soru 23Soru →

A right circular cylindrical tank with a base radius of $4\text{ inches}$ contains water. A solid metal sphere with a radius of $3\text{ inches}$ is completely submerged in the water, causing the water level to rise. If no water overflows from the tank, what is the increase, in inches, in the water level of the tank?

0.5625

9.00

2.25

6.75

Cevabı ve açıklamayı göster

Cevap: 2.25

Cevap

2.25

The correct answer is 2.25. The volume of the submerged sphere is calculated using the formula

V = \frac{4}{3}\pi r^3

. With a radius of

3\text{ inches}

, the volume is

V = \frac{4}{3}\pi (3)^3 = 36\pi\text{ cubic inches}

. The volume of the displaced water is represented by a cylinder of radius

4\text{ inches}

and height

h

, which represents the increase in water level. The volume of this displaced cylinder is

V = \pi (4)^2 h = 16\pi h

. Since the volume of the displaced water is equal to the volume of the submerged sphere, we set them equal:

16\pi h = 36\pi

. Dividing both sides by

16\pi

yields

h = 2.25

.

Adım Adım Çözüm

1

Calculate the volume of the solid metal sphere using the formula

V = \frac{4}{3}\pi r^3

with radius

r = 3\text{ inches}

.

V_{\text{sphere}} = \frac{4}{3}\pi (3)^3 = 36\pi\text{ cubic inches}

The volume of the sphere represents the total volume of water that will be displaced when the sphere is completely submerged.

2

Express the volume of the displaced water in the cylinder as a function of the rise in water level,

h

.

V_{\text{displaced}} = \pi (4)^2 h = 16\pi h\text{ cubic inches}

The displaced water forms a cylindrical shape with the same base radius as the tank (

4\text{ inches}

) and a height equal to the change in the water level (

h

).

3

Equate the volume of the displaced water to the volume of the sphere and solve for

h

.

16\pi h = 36\pi \Rightarrow h = \frac{36}{16} = 2.25\text{ inches}

By Archimedes' principle, the volume of the displaced water must equal the volume of the fully submerged solid sphere.

Anahtar Kavram

Equating the volume of a submerged object to the volume of displaced liquid in a cylindrical container.

Tahmini Süre:2m 0s

Soru 24Soru →

A right square pyramid has a height of $9\text{ centimeters}$ and a square base with a side length of $5\text{ centimeters}$ . What is the volume, in cubic centimeters, of the pyramid?

15

45

75

225

Cevabı ve açıklamayı göster

Cevap: 75

Cevap

75

The volume of a right square pyramid is given by the formula

V = \frac{1}{3}Bh

, where

B

is the area of the base and

h

is the height. Since the base is a square with side length

5\text{ centimeters}

, its area is

B = 5^2 = 25\text{ square centimeters}

. Substituting the base area and the height of

9\text{ centimeters}

into the formula gives

V = \frac{1}{3} \times 25 \times 9 = 75\text{ cubic centimeters}

.

Adım Adım Çözüm

1

Calculate the area of the square base (

B

).

B = 5^2 = 25\text{ square centimeters}

The base is a square with side length

5\text{ centimeters}

, and the area of a square is calculated by squaring its side length.

2

Apply the volume formula for a pyramid:

V = \frac{1}{3}Bh

.

V = \frac{1}{3} \times 25 \times 9 = 75\text{ cubic centimeters}

Substituting the base area of

25\text{ square centimeters}

and the height of

9\text{ centimeters}

into the pyramid volume formula yields the volume.

Anahtar Kavram

The volume of a pyramid is calculated using the formula

V = \frac{1}{3}Bh

, where

B

is the area of the base and

h

is the height.

Tahmini Süre:45s

Soru 25Soru →

A solid metal right circular cylinder has a height of $h$ and a base radius of $r$ , where $h > 2r$ . From one flat end of the cylinder, a hemisphere of radius $r$ is carved out. From the other flat end, a right circular cone of base radius $r$ and height $h - 2r$ is carved out. If the volume of the remaining solid is equal to the volume of a sphere with radius $R$ , which of the following equations represents $R$ in terms of $r$ and $h$ ?

R = \sqrt[3]{\frac{r^2 h}{2}}

R = \sqrt{\frac{r^2 h}{2}}

R = \sqrt[3]{r^2 h}

R = \frac{r^2 h}{2}

Cevabı ve açıklamayı göster

Cevap: $R = \sqrt[3]{\frac{r^2 h}{2}}$

Cevap

The correct equation is

R = \sqrt[3]{\frac{r^2 h}{2}}

The volume of the cylinder is

\pi r^2 h

. Subtracting the hemisphere's volume (

\frac{2}{3}\pi r^3

) and the cone's volume (

\frac{1}{3}\pi r^2(h - 2r) = \frac{1}{3}\pi r^2 h - \frac{2}{3}\pi r^3

) leaves a remaining volume of

\frac{2}{3}\pi r^2 h

. Setting this equal to the volume of a sphere of radius

R

(

\frac{4}{3}\pi R^3

) gives

\frac{4}{3}\pi R^3 = \frac{2}{3}\pi r^2 h

. Simplifying this equation yields

R^3 = \frac{r^2 h}{2}

, and taking the cube root gives the correct relation.

Adım Adım Çözüm

1

Write the formula for the volume of the original cylinder.

V_{\text{cylinder}} = \pi r^2 h

The volume of a cylinder with radius

r

and height

h

is given by the formula

\pi r^2 h

.

2

Write the volume formulas for the carved-out shapes.

V_{\text{hemisphere}} = \frac{2}{3}\pi r^3

and

V_{\text{cone}} = \frac{1}{3}\pi r^2 (h - 2r)

A hemisphere is half a sphere, so its volume is

\frac{2}{3}\pi r^3

. The volume of a cone with radius

r

and height

h - 2r

is

\frac{1}{3}\pi r^2 (h - 2r)

.

3

Subtract the volumes of the carved-out shapes from the cylinder's volume.

V_{\text{remaining}} = \pi r^2 h - \frac{2}{3}\pi r^3 - \frac{1}{3}\pi r^2 (h - 2r) = \frac{2}{3}\pi r^2 h

Expanding the cone's volume gives

\frac{1}{3}\pi r^2 h - \frac{2}{3}\pi r^3

. Subtracting this and the hemisphere's volume from the cylinder's volume simplifies to

\frac{2}{3}\pi r^2 h

because the

\pi r^3

terms cancel out.

4

Equate the remaining volume to the volume of a sphere with radius

R

and solve for

R

.

\frac{4}{3}\pi R^3 = \frac{2}{3}\pi r^2 h \implies R^3 = \frac{r^2 h}{2} \implies R = \sqrt[3]{\frac{r^2 h}{2}}

Equating the remaining volume to the volume of a sphere of radius

R

and dividing both sides by

\frac{2}{3}\pi

gives

2 R^3 = r^2 h

. Dividing by 2 and taking the cube root isolates

R

.

Anahtar Kavram

Calculating and comparing volumes of combined geometric solids and algebraically isolating variables.

Tahmini Süre:3m 0s

Soru 26Soru →

Right circular Cylinder $A$ has a base radius of $r$ and a height of $h$ . Right circular Cylinder $B$ has a base radius that is $3$ times the base radius of Cylinder $A$ , and a height that is $\frac{1}{2}$ the height of Cylinder $A$ . What is the ratio of the volume of Cylinder $B$ to the volume of Cylinder $A$ ?

\frac{3}{2}

\frac{9}{4}

\frac{9}{2}

18

Cevabı ve açıklamayı göster

Cevap: $\frac{9}{2}$

Cevap

The ratio of the volume of Cylinder

B

to the volume of Cylinder

A

is

\frac{9}{2}

.

The volume of a cylinder is given by the formula

V = \pi r^2 h

. For Cylinder

A

, the volume is

V_A = \pi r^2 h

. For Cylinder

B

, the radius is

3r

and the height is

\frac{1}{2}h

. Substituting these values gives

V_B = \pi (3r)^2 \left(\frac{1}{2}h\right) = \frac{9}{2}\pi r^2 h

. The ratio of the volume of Cylinder

B

to the volume of Cylinder

A

is therefore the ratio of nine to two.

Adım Adım Çözüm

1

Write the volume formula for Cylinder

A

in terms of its radius

r

and height

h

.

V_A = \pi r^2 h

To establish a baseline volume for Cylinder

A

using the standard formula for the volume of a right circular cylinder.

2

Express the radius and height of Cylinder

B

in terms of

r

and

h

, and substitute them into the volume formula.

The radius of Cylinder

B

is

3r

and the height is

\frac{1}{2}h

. Thus,

V_B = \pi (3r)^2 \left(\frac{1}{2}h\right) = \pi (9r^2) \left(\frac{1}{2}h\right) = \frac{9}{2}\pi r^2 h

.

To find the volume of Cylinder

B

expressed in terms of the variables

r

and

h

.

3

Divide the volume of Cylinder

B

by the volume of Cylinder

A

to find the ratio.

\frac{V_B}{V_A} = \frac{\frac{9}{2}\pi r^2 h}{\pi r^2 h} = \frac{9}{2}

To calculate the ratio of the volume of Cylinder

B

to Cylinder

A

.

Anahtar Kavram

Dimensional scaling of the volume of a cylinder

Soru 27Soru →

A square picture has an area of $16$ square inches. If the side length of the picture is doubled, what is the area, in square inches, of the new picture?

8

32

64

128

Cevabı ve açıklamayı göster

Cevap: $64$

Cevap

64

square inches

The area of a square with side length

s

is given by

s^2

. Since the original area is

16

square inches, the original side length is

\sqrt{16} = 4

inches. Doubling this side length gives a new side length of

8

inches. Therefore, the area of the new square is

8^2 = 64

square inches. Alternatively, when all linear dimensions of a two-dimensional shape are multiplied by a scale factor

k

, the area is multiplied by

k^2

. Since the side length is doubled (

k = 2

), the area is multiplied by

2^2 = 4

, resulting in

16 \times 4 = 64

square inches.

Adım Adım Çözüm

1

Determine the side length of the original square.

The side length is

\sqrt{16} = 4

inches.

The area of a square is given by

A = s^2

, where

s

is the side length. So,

s = \sqrt{A}

.

2

Calculate the side length of the new square after doubling.

The new side length is

4 \times 2 = 8

inches.

The problem states that the side length is doubled.

3

Calculate the area of the new square.

The new area is

8^2 = 64

square inches.

The area of the new square is the square of its new side length (

8 \text{ inches} \times 8 \text{ inches}

).

Anahtar Kavram

When a two-dimensional shape is scaled by a factor of

k

, its area is scaled by a factor of

k^2

.

Soru 28Soru →

A solid rectangular prism has a volume of $192$ cubic inches. The ratio of the length of the prism to its width is $3:1$ , and the height of the prism is $4$ inches. What is the surface area, in square inches, of the prism?

Cevabı ve açıklamayı göster

Cevap: 224

Cevap

The surface area of the prism is 224 square inches.

The correct answer is 224. By using the volume formula

V = lwh

with a volume of

192

and a height of

4

, the product of the length and width of the base is

lw = 48

. Given that the ratio of the length to the width is

3:1

, the length can be written as

l = 3w

. Substituting this expression into the product equation gives

3w(w) = 48

, or

3w^2 = 48

. Dividing both sides by 3 yields

w^2 = 16

, which means the width of the prism is

4

inches. Since

l = 3w

, the length is

12

inches. The surface area of the prism can be found using the formula

SA = 2(lw + lh + wh)

. Substituting the dimensions

l = 12

,

w = 4

, and

h = 4

yields

SA = 2(12 \cdot 4 + 12 \cdot 4 + 4 \cdot 4) = 2(48 + 48 + 16) = 2(112) = 224

square inches.

Adım Adım Çözüm

1

Use the volume formula for a rectangular prism,

V = lwh

, and substitute the given volume of

192

and height of

4

.

192 = l \cdot w \cdot 4

, which simplifies to

l \cdot w = 48

.

To find the product of the length and width of the base of the prism.

2

Express the length in terms of the width using the ratio of

3:1

.

l = 3w

.

The ratio of the length to the width is given as 3 to 1.

3

Substitute

l = 3w

into the equation

l \cdot w = 48

and solve for

w

.

3w \cdot w = 48 \implies 3w^2 = 48 \implies w^2 = 16 \implies w = 4

.

To solve for the width of the rectangular prism.

4

Find the length of the rectangular prism.

l = 3(4) = 12

inches.

Since the length is 3 times the width and the width is 4 inches, the length must be 12 inches.

5

Use the surface area formula

SA = 2(lw + lh + wh)

with the dimensions

l = 12

,

w = 4

, and

h = 4

.

SA = 2(12 \cdot 4 + 12 \cdot 4 + 4 \cdot 4) = 2(48 + 48 + 16) = 2(112) = 224

.

To calculate the total surface area of the prism.

Anahtar Kavram

Volume and Surface Area of Rectangular Prisms

Soru 29Soru →

A right circular cylinder has a height of $12\text{ centimeters}$ . If the total surface area of the cylinder is $170\pi\text{ square centimeters}$ , what is the volume, in cubic centimeters, of the cylinder?

120\pi

300\pi

2040\pi

3468\pi

Cevabı ve açıklamayı göster

Cevap: $300\pi$

Cevap

The volume of the cylinder is

300\pi

cubic centimeters.

The correct answer is the volume of

300\pi

cubic centimeters. We find this by using the cylinder's surface area equation

2\pi r h + 2\pi r^2 = 170\pi

with

h = 12

, which reduces to

r^2 + 12r - 85 = 0

. Factoring this equation gives the radius

r = 5

. Plugging

r = 5

and

h = 12

into the volume formula

V = \pi r^2 h

gives

300\pi

.

Adım Adım Çözüm

1

Set up the formula for the total surface area of a cylinder and plug in the given values.

2\pi r h + 2\pi r^2 = 170\pi

, where

h = 12

. This simplifies to

2\pi r(12) + 2\pi r^2 = 170\pi

, or

24\pi r + 2\pi r^2 = 170\pi

.

We need to find the radius of the cylinder to calculate its volume.

2

Divide the entire equation by

2\pi

to simplify it into a standard quadratic form.

r^2 + 12r - 85 = 0

.

Simplifying the equation makes it easier to solve for the radius

r

.

3

Factor the quadratic equation to find the positive value of the radius

r

.

(r + 17)(r - 5) = 0

, which gives

r = 5

since the radius must be positive (

r > 0

).

Factoring allows us to solve for the variable

r

.

4

Use the radius

r = 5

and height

h = 12

to compute the volume of the cylinder.

V = \pi r^2 h = \pi (5^2)(12) = 25 \times 12 \pi = 300\pi

cubic centimeters.

The volume formula for a cylinder is

V = \pi r^2 h

.

Anahtar Kavram

Calculating volume from total surface area for a cylinder by solving a quadratic relation.

Soru 30Soru →

A trapezoid has parallel bases of length $6$ centimeters and $10$ centimeters, and a height of $5$ centimeters. What is the area, in square centimeters, of the trapezoid?

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

The area of the trapezoid is

40

square centimeters.

The area of a trapezoid is calculated using the formula

A = \frac{a + b}{2} h

, where

a

and

b

are the lengths of the parallel bases and

h

is the height. Substituting the given values

a = 6

,

b = 10

, and

h = 5

into the formula yields

A = \frac{6 + 10}{2} \times 5 = 8 \times 5 = 40

.

Adım Adım Çözüm

1

Recall the area formula for a trapezoid.

A = \frac{a + b}{2}h

The area of a trapezoid is the average of its parallel bases multiplied by its height.

2

Substitute the base values of

6

and

10

, and the height value of

5

into the formula.

A = \frac{6 + 10}{2} \times 5

This sets up the calculation with the given measurements.

3

Simplify the expression to find the final area.

A = 40

Adding the bases gives

16

, dividing by

2

gives

8

, and multiplying by the height of

5

results in

40

.

Anahtar Kavram

Area of a Trapezoid

Tahmini Süre:45s

Soru 31Soru →

In right triangle $ABC$ , the measure of angle $B$ is $90^\circ$ . If $\sin(A) = \frac{5}{13}$ , what is the value of $\cos(C)$ ?

\frac{5}{13}

\frac{12}{13}

\frac{13}{5}

\frac{8}{13}

Cevabı ve açıklamayı göster

Cevap: $\frac{5}{13}$

Cevap

\frac{5}{13}

The correct answer is

\frac{5}{13}

. In any right triangle

ABC

where the right angle is at

B

, the two acute angles

A

and

C

are complementary, meaning

A + C = 90^\circ

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

\sin(A) = \cos(90^\circ - A) = \cos(C)

. Given that

\sin(A) = \frac{5}{13}

, the value of

\cos(C)

must also be

\frac{5}{13}

.

Adım Adım Çözüm

1

Identify the relationship between the acute angles in a right triangle.

Since the measure of angle

B

is

90^\circ

, the sum of the measures of angles

A

and

C

must be

90^\circ

. Thus, angles

A

and

C

are complementary.

The sum of angles in any triangle is

180^\circ

.

2

Apply the co-function identity for complementary angles.

For any two complementary angles

A

and

C

, the identity

\cos(C) = \sin(A)

holds true.

The sine of an angle is the ratio of the opposite side to the hypotenuse, which is the same as the ratio of the adjacent side of its complement to the hypotenuse (cosine of the complement).

3

Substitute the given value to find

\cos(C)

.

Since

\sin(A) = \frac{5}{13}

, it follows that

\cos(C) = \frac{5}{13}

.

Direct substitution into the identity

\cos(C) = \sin(A)

.

Anahtar Kavram

Co-function identities for complementary angles in a right triangle

Soru 32Soru →

A right circular cylindrical container with a base radius of $R$ inches is partially filled with water. A solid metal sphere with a radius of $r$ inches is placed into the container and becomes completely submerged, causing the water level to rise by $1.5$ inches without any water overflowing. The sphere is then removed, and a solid right circular cone with a base radius of $r$ inches and a height of $12$ inches is placed vertex-down into the container. When the cone is completely submerged, the water level is $1$ inch higher than the container's original water level. What is the value of $R$ , in inches?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The radius of the cylinder,

R

, is

9

inches.

The volume of a submerged solid is equal to the volume of the cylinder of water it displaces. By setting up equations for the sphere and the cone, we get

\frac{4}{3}r^3 = 1.5 R^2

and

4r^2 = R^2

. Substituting the second equation into the first yields

r = 4.5

, which then gives

R = 2r = 9

.

Adım Adım Çözüm

1

Equate the volume of the sphere to the volume of water it displaces in the cylinder.

\frac{4}{3}\pi r^3 = 1.5\pi R^2 \implies \frac{4}{3}r^3 = 1.5 R^2

The volume of a submerged solid equals the volume of the fluid it displaces. The displaced fluid takes the shape of a cylinder of radius

R

and height equal to the water level rise (

1.5

inches).

2

Equate the volume of the cone to the volume of water it displaces in the cylinder.

\frac{1}{3}\pi r^2 (12) = 1.0\pi R^2 \implies 4r^2 = R^2

Similarly, the volume of the cone is equal to the volume of a cylinder of radius

R

and height equal to the water level rise (

1

inch).

3

Solve the system of equations by substituting

R^2

into the sphere's displacement equation.

\frac{4}{3}r^3 = 1.5(4r^2) \implies \frac{4}{3}r^3 = 6r^2 \implies r = 4.5

By replacing

R^2

with

4r^2

, we reduce the system of equations to a single equation containing only

r

. Since

r \neq 0

, we can divide by

r^2

to solve for

r

directly.

4

Calculate the value of

R

from the relationship between

R

and

r

.

R = 2r = 2(4.5) = 9

Since

R^2 = 4r^2

and radii must be positive quantities,

R = 2r

.

Anahtar Kavram

Using water displacement to relate the volumes of three-dimensional geometric solids (cylinders, spheres, cones) and solving non-linear systems of equations.

Soru 33Soru →

A container in the shape of a right circular cylinder with a base radius of $4$ inches and a height of $9$ inches is completely filled with water. All of the water from this cylinder is poured into an empty container in the shape of a rectangular prism. The rectangular prism container has a square base of side length $6$ inches and a height of $15$ inches. If the height of the water in the rectangular prism container is $k\pi$ inches, what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

k

is

4

.

To find the height of the water in the rectangular prism, we first calculate the volume of the water using the cylinder's volume formula,

V = \pi r^2 h

. With a radius of

4

inches and a height of

9

inches, the volume is

\pi (4)^2 (9) = 144\pi

cubic inches. When this water is poured into the rectangular prism, the volume of the water can also be represented as the area of the base times the height of the water:

V = \text{Base Area} \times h_w = 6^2 \times h_w = 36 h_w

. Equating the two volumes gives

36 h_w = 144\pi

, which simplifies to

h_w = 4\pi

inches. Since the height of the water in the prism is

k\pi

inches, the value of

k

is

4

.

Adım Adım Çözüm

1

Calculate the volume of the water in the filled cylinder using the formula

V = \pi r^2 h

.

The volume of the water is

144\pi

cubic inches.

Since the cylinder is completely filled, the volume of the water is equal to the volume of the cylinder with base radius

4

inches and height

9

inches.

2

Calculate the area of the square base of the rectangular prism container.

The base area is

36

square inches.

The base of the prism is a square with side length

6

inches, so its area is

6 \times 6 = 36

square inches.

3

Set the volume of the water in the rectangular prism equal to the volume of the water from the cylinder, and solve for the water height

h_w

.

36 \times h_w = 144\pi \implies h_w = 4\pi

inches.

Pouring the water into the prism container does not change its volume, which remains

144\pi

cubic inches. The volume of a prism is the base area times its height.

4

Compare the height of the water

4\pi

to the expression

k\pi

to find

k

.

k = 4

.

Since the height of the water is represented as

k\pi

inches and we calculated it to be

4\pi

inches,

k

must equal

4

.

Anahtar Kavram

Equating the volumes of a right circular cylinder and a rectangular prism to solve for an unknown dimension.

Soru 34Soru →

A solid metal right circular cylinder has a base radius of $12\text{ centimeters}$ and a height of $3\text{ centimeters}$ . A cone with the same base and height as the cylinder is carved out and removed. The remaining metal is melted and recast into a solid sphere. What is the radius, in centimeters, of the sphere?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The radius of the sphere is 6 centimeters.

The volume of a cylinder is

V = \pi r^2 h = \pi (12)^2 (3) = 432\pi

. The volume of the carved-out cone is

V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (12)^2 (3) = 144\pi

. Subtracting the cone's volume from the cylinder's volume yields the remaining metal volume:

432\pi - 144\pi = 288\pi

. The volume of the recast sphere is

\frac{4}{3}\pi R^3 = 288\pi

. Dividing by

\pi

and multiplying by

\frac{3}{4}

gives

R^3 = 216

. Taking the cube root of both sides gives the radius

R = 6

.

Adım Adım Çözüm

1

Calculate the volume of the original cylinder.

432\pi\text{ cubic centimeters}

This determines the starting volume of the solid metal block before any material is removed.

2

Calculate the volume of the cone that is carved out.

144\pi\text{ cubic centimeters}

This determines how much metal is discarded from the cylinder.

3

Subtract the cone's volume from the cylinder's volume.

288\pi\text{ cubic centimeters}

This gives the volume of the remaining metal that will be melted and recast.

4

Equate the remaining volume to the volume formula of a sphere and solve for the radius.

6\text{ centimeters}

This yields the radius of the newly formed sphere.

Anahtar Kavram

Volume of composite solids and conservation of volume during recasting

Soru 35Soru →

A solid right circular cone has a base radius of $6\text{ centimeters}$ and a height of $8\text{ centimeters}$ . A sphere is inscribed inside the cone such that it is tangent to the base of the cone and tangent to the lateral surface of the cone. What is the ratio of the volume of the sphere to the volume of the cone?

\frac{1}{9}

\frac{3}{8}

\frac{1}{8}

\frac{8}{9}

Cevabı ve açıklamayı göster

Cevap: $\frac{3}{8}$

Cevap

The ratio of the volume of the sphere to the volume of the cone is

\frac{3}{8}

.

The correct answer is found by first determining the radius of the inscribed sphere. By analyzing the 2D cross-section of the cone, which is an isosceles triangle with a base of

12

and a height of

8

, the slant height is found to be

10

using the Pythagorean theorem. The radius of the inscribed circle (the sphere's radius) is calculated using the area and semiperimeter of the triangle, yielding

R = 3

. The volume of the sphere is

36\pi

and the volume of the cone is

96\pi

. Dividing the sphere's volume by the cone's volume gives the ratio of

\frac{3}{8}

.

Adım Adım Çözüm

1

Find the slant height of the cone.

Slant height

L = 10\text{ centimeters}

.

The vertical cross-section of the cone is an isosceles triangle with base

2r = 12

and height

h = 8

. The slant height is the hypotenuse of the right triangle formed by the radius, height, and slant height:

L = \sqrt{6^2 + 8^2} = 10

.

2

Calculate the radius of the inscribed sphere.

Sphere radius

R = 3\text{ centimeters}

.

The sphere's radius is the inradius of the cross-sectional isosceles triangle. The area of this triangle is

A = \frac{1}{2} \times 12 \times 8 = 48

, and its semiperimeter is

s = \frac{12 + 10 + 10}{2} = 16

. Using the inradius formula

R = \frac{A}{s}

, we get

R = \frac{48}{16} = 3

.

3

Calculate the volume of the cone.

Volume of the cone

V_{\text{cone}} = 96\pi\text{ cubic centimeters}

.

Using the volume of a cone formula,

V_{\text{cone}} = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (6^2)(8) = 96\pi

.

4

Calculate the volume of the sphere.

Volume of the sphere

V_{\text{sphere}} = 36\pi\text{ cubic centimeters}

.

Using the volume of a sphere formula,

V_{\text{sphere}} = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (3^3) = 36\pi

.

5

Compute the ratio of the volume of the sphere to the volume of the cone.

Ratio

= \frac{3}{8}

.

Divide the volume of the sphere by the volume of the cone:

\frac{36\pi}{96\pi} = \frac{36}{96} = \frac{3}{8}

.

Anahtar Kavram

Calculating the volume of spheres and cones, and using 2D cross-sections to find missing geometric dimensions of inscribed solids.

Soru 36Soru →

A circular rug has an area of $16\pi$ square feet. If the radius of the rug is tripled, what is the area, in square feet, of the new rug?

32\pi

48\pi

144\pi

64\pi

Cevabı ve açıklamayı göster

Cevap: $144\pi$

Cevap

The area of the new rug is

144\pi

square feet.

The area of a circle scales with the square of its radius. If the radius is tripled, the area is multiplied by

3^2 = 9

. Since the original area is

16\pi

square feet, the area of the new rug is

16\pi \times 9 = 144\pi

square feet.

Adım Adım Çözüm

1

Determine the relationship between the linear scale factor and the area scale factor of a circle.

If the radius of a circle is multiplied by a scale factor

k

, the area of the circle is multiplied by

k^2

.

The area of a circle is proportional to the square of its radius (

A = \pi r^2

).

2

Find the area scale factor when the radius is tripled.

Since the radius is multiplied by

3

, the area scale factor is

3^2 = 9

.

Squaring the linear scale factor of 3 gives the quadratic scale factor for the area.

3

Calculate the area of the new rug.

The new area is

16\pi \times 9 = 144\pi

square feet.

Multiplying the original area of

16\pi

square feet by the area scale factor of 9 yields the final scaled area.

Anahtar Kavram

Area scaling of two-dimensional shapes

Tahmini Süre:1m 0s

Soru 37Soru →

Two straight support beams on a bridge intersect at a single point. One of the angles formed by their intersection measures $74^\circ$ . An adjacent angle along the straight line of one of the beams has a measure of $(2x + 16)^\circ$ . What is the value of $x$ ?

29

45

61

90

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

45

Since the two angles form a linear pair along a straight support beam, their measures must sum to

180^\circ

. We can write the equation as

74 + (2x + 16) = 180

. Combining like terms gives

2x + 90 = 180

. Subtracting

90

from both sides yields

2x = 90

. Finally, dividing by

2

gives

x = 45

.

Adım Adım Çözüm

1

Set up the equation using the property of supplementary angles.

74 + (2x + 16) = 180

Adjacent angles on a straight line form a linear pair, which means they are supplementary and sum to

180^\circ

.

2

Combine the constant terms on the left side of the equation.

2x + 90 = 180

Adding

74

and

16

simplifies the constant terms to

90

.

3

Isolate the variable term by subtracting 90 from both sides of the equation.

2x = 90

Subtracting

90

from

180

leaves

90

on the right side.

4

Solve for x by dividing both sides of the equation by 2.

x = 45

Dividing

2x

and

90

by

2

isolates the variable

x

.

Anahtar Kavram

Adjacent angles on a straight line are supplementary and add up to 180 degrees.

Tahmini Süre:45s

Soru 38Soru →

If $\sin(x^\circ) = \cos(38^\circ)$ , where $0 < x < 90$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 52

Cevap

The value of

x

is 52.

Applying the co-function identity

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

allows us to set the argument of the sine function as the complement of the cosine's argument. Thus,

x = 90 - 38 = 52

.

Adım Adım Çözüm

1

Identify the relevant trigonometric identity.

The co-function identity

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

is appropriate here.

We need to relate the sine of one angle to the cosine of another angle.

2

Equate the angle arguments using the identity.

\sin(x^\circ) = \cos(90^\circ - x^\circ) = \cos(38^\circ)

, which implies

90 - x = 38

.

For acute angles, if their cosine values are equal, the angles themselves must be equal.

3

Solve the linear equation for

x

.

x = 90 - 38 = 52

.

Isolating

x

gives the final answer.

Anahtar Kavram

Co-function identities relate the sine of an angle to the cosine of its complement:

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

.

Soru 39Soru →

In right triangle $XYZ$ , the measure of angle $Y$ is $90^\circ$ , $XY = 12$ , and $YZ = 5$ . What is the value of $\cos(X) - \sin(X)$ ?

\frac{7}{13}

\frac{17}{13}

-\frac{7}{13}

\frac{7}{12}

Cevabı ve açıklamayı göster

Cevap: $\frac{7}{13}$

Cevap

\frac{7}{13}

The length of the hypotenuse

XZ

is first found using the Pythagorean theorem:

XZ = \sqrt{12^2 + 5^2} = 13

. Using the definitions of the trigonometric ratios,

\cos(X) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}

and

\sin(X) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}

. The difference is

\cos(X) - \sin(X) = \frac{12}{13} - \frac{5}{13} = \frac{7}{13}

.

Adım Adım Çözüm

1

Find the length of the hypotenuse

XZ

using the Pythagorean theorem.

XZ = \sqrt{XY^2 + YZ^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13

.

The hypotenuse is needed as the denominator for the sine and cosine ratios.

2

Determine the values of

\cos(X)

and

\sin(X)

.

\cos(X) = \frac{XY}{XZ} = \frac{12}{13}

and

\sin(X) = \frac{YZ}{XZ} = \frac{5}{13}

.

By definition,

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

and

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

.

3

Calculate the difference

\cos(X) - \sin(X)

.

\cos(X) - \sin(X) = \frac{12}{13} - \frac{5}{13} = \frac{7}{13}

.

This is the final subtraction requested by the question.

Anahtar Kavram

Calculating trigonometric ratios in a right triangle and applying basic operations.

Soru 40Soru →

A triangular banner has an area of $30$ square feet. If the height of the banner is $5$ feet, what is the length, in feet, of the base of the banner?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The length of the base of the banner is

12

feet.

To find the base of the triangular banner, apply the formula for the area of a triangle:

A = \frac{1}{2} b h

. Substituting

30

for the area

A

and

5

for the height

h

yields

30 = \frac{1}{2} b (5)

, or

30 = 2.5b

. Dividing both sides of the equation by

2.5

gives

b = 12

. Therefore, the length of the base of the banner is

12

feet.

Adım Adım Çözüm

1

Recall the formula for the area of a triangle.

A = \frac{1}{2} b h

The area of a triangle is equal to half the product of its base and height.

2

Substitute the given values into the area formula.

30 = \frac{1}{2} \times b \times 5

The problem provides the area (

30

square feet) and the height (

5

feet).

3

Solve for the base

b

.

b = 12

Multiply both sides of the equation by

2

to clear the fraction, giving

60 = 5b

. Then, divide both sides by

5

to find that

b = 12

.

Anahtar Kavram

Area of a triangle

Geometry and Trigonometry

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