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HiSET: Math : Use volume formulas to solve problems

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Example Questions

Example Question #1 : Cones

A cylinder has volume 120. A cone with base the same size as a base of the cylinder has the same height. Give the volume of the cone.

Possible Answers:

Insufficient information is given to answer the question.

Correct answer:

Explanation:

A cone with the same height as a given cylinder and a base the same radius as those of that cylinder has as its volume one-third that of the cylinder. That makes the volume of the cone one-third of 120, or

$V = \frac{1}{3}(120) = 40$

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Example Question #3 : Cones

A right cone has height 10 and slant height 20. Which of the following correctly gives its volume? (Round to the nearest whole number).

Possible Answers:

942

3,142

1,814

2,094

1,885

Correct answer:

3,142

Explanation:

The volume of a cone, given radius and height , can be calculated using the formula

$V = \frac{1}{3} \pi r^{2}h$ .

We are given that h = 10 , but we are not given the value of . We are given that l = 20 , and since the cone is a right cone, its radius, height, and slant height can be related using the Pythagorean relation

$r^{2}+ h^{2}= l^{2}$ .

Substituting 10 for and 20 for , we can find :

$r^{2}+ 10^{2} = 20^{2}$

$r^{2}+ 100=400$

$r^{2} + 100- 100 =400 - 100$

$r^{2} =300$ , which is what we need in the formula.

Now substitute in the volume formula:

$V = \frac{1}{3} \pi r^{2}h$

$\approx \frac{1}{3}(3.1416 )(300)(10)$

$\approx 3141.6$

This rounds to 3,142

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Example Question #4 : Cones

A cone has as its base a circle whose radius is twice that of a base of a given cylinder; its height is 20% greater than that of the cylinder. Which of the following is true of the volume of the cone?

Possible Answers:

The volume of the cone is equal than that of the cylinder.

The volume of the cone is 20% greater than that of the cylinder.

The volume of the cone is 60% less than that of the cylinder.

The volume of the cone is 60% greater than that of the cylinder.

The volume of the cone is 20% less than that of the cylinder.

Correct answer:

The volume of the cone is 60% greater than that of the cylinder.

Explanation:

The volume of a cylinder with a base of radius and with height is

$V = \pi r^{2}h$

The cone has radius twice that of the cylinder, which is . Its height is 20% greater than, or 120% of, that of the cylinder, which is equal to 1.2h .

The volume of a cone with a base of radius and height is

$V = \frac{1}{3}\pi r'^{2}h'$

Set r' = 2r and h'= 1.2h :

$V = \frac{1}{3}\pi (2r)^{2} \left (1.2h \right )$

$= \frac{1}{3}\pi (2^{2} r^{2} )\left (1.2h \right )$

$= \frac{1}{3}\pi (4 r^{2} )\left (1.2h \right )$

$= \left (\frac{1}{3} \cdot 4 \cdot 1.2 \right ) \pi r^{2} h$

$= 1.6 \pi r^{2} h$

Substitute for $\pi r^{2}h$ :

V' =1.6V .

This means that the volume of the cone is $1.6 = 1.6 \times 100\% = 160\%$ times that of the cylinder—or 60% greater.

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Example Question #5 : Cones

About the -axis, rotate the triangle whose sides are along the -axis, the -axis, and the line of the equation

2x + y = 8 .

Give the volume of the solid of revolution formed.

Possible Answers:

$48 \pi$

$96 \pi$

$\frac{128}{3} \pi$

$\frac{256}{3} \pi$

None of the other choices gives the correct response.

Correct answer:

$\frac{256}{3} \pi$

Explanation:

The vertices of the triangle are the points of intersection of the three lines. The -axis and the -axis meet at the origin (0,0) .

The point of intersection of the -axis and the graph of 2x + y = 8 —the -intercept of the latter—can be found by substituting 0 for :

2x + y = 8

2x + 0= 8

2x = 8

Divide both sides by 2 to isolate :

$2x \div 2 = 8 \div 2$

x = 4

The point of intersection is at (4,0) .

Similarly, the point of intersection of the -axis and the graph of 2x + y = 8 —the -intercept of the latter—can be found by substituting 0 for :

2x + y = 8

2 (0) + y = 8

0 + y= 8

y = 8

The point of intersection is at (0,8) .

The three vertices of the triangle are at the origin, (4,0) , and (0,8) . When this triangle is rotated about the -axis, the resulting solid of revolution is a cone whose base has radius r= 8 , and which has height h= 4 . Substitute these values into the formula for the volume of a cone:

$V = \frac{1}{3} \pi r^{2}h$

$= \frac{1}{3} \pi (8^{2}) (4)$

$= \frac{1}{3}\cdot 64 \cdot 4 \cdot \pi$

$= \frac{256}{3} \pi$

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Example Question #2 : Cones

A right cone has height 20; its base has radius 10. Which of the following correctly gives its volume? (Round to the nearest whole number).

Possible Answers:

1,885

942

1,814

2,094

3,142

Correct answer:

2,094

Explanation:

The volume of a cone, given radius and height , can be calculated using the formula

$V = \frac{1}{3} \pi r^{2}h$ .

We are given that r = 10 and h= 20 , so we can substitute and calculate:
$V = \frac{1}{3} \pi (10^{2} )(20)$

$\approx \frac{1}{3} \cdot 3.1416 \cdot 100 \cdot 20$

$\approx 2094.4$

To the nearest whole, this is 2,094.

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Example Question #7 : Cones

A cone has as its base a circle whose radius is 20% greater that of a base of the cylinder; its height is twice that of the cylinder. Which of the following is true of the volume of the cone?

Possible Answers:

The volume of the cone is 4% less than that of the cylinder.

The volume of the cone is 52% less than that of the cylinder.

The volume of the cone is equal than that of the cylinder.

The volume of the cone is 52% greater than that of the cylinder.

The volume of the cone is 4% greater than that of the cylinder.

Correct answer:

The volume of the cone is 4% less than that of the cylinder.

Explanation:

The volume of a cylinder with a base of radius and with height is

$V = \pi r^{2}h$

The cone has height twice that of the cylinder, which is . Its radius is 20% greater than, or 120% of, that of the cylinder, which is equal to 1.2r .

The volume of a cone with a base of radius and height is

$V = \frac{1}{3}\pi r'^{2}h'$

Set r'=1.2r and h' = 2h :

$V = \frac{1}{3}\pi \left ( 1.2r\right )^{2}(2h)$

$= \frac{1}{3}\pi \left ( 1.2^{2} \right )r^{2} (2h)$

$= \frac{1}{3} \left ( 1.44 \right ) (2) \cdot \pi r^{2} h$

$=0.96 \cdot \pi r^{2} h$

Substitute for $\pi r^{2}h$ :

V'= 0.96 V

This means that the volume of the cone is $0.96= 0.96 \times 100\% = 96\%$ that of the cylinder—or $(100 - 96) \% = 4 \%$ less.

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Example Question #11 : Cones

About the y-axis, rotate the triangle formed with the x-axis, the y-axis, and the line of the equation

2x + y = 8 .

Give the volume of the solid of revolution formed.

Possible Answers:

$\frac{128}{3} \pi$

None of the other choices gives the correct response.

$96 \pi$

$\frac{256}{3} \pi$

$48 \pi$

Correct answer:

$\frac{128}{3} \pi$

Explanation:

The vertices of the triangle are the points of intersection of the three lines. The -axis and the -axis meet at the origin (0,0) .

The point of intersection of the -axis and the graph of 2x + y = 8 - the -intercept of the latter - can be found by substituting 0 for :

2x + y = 8

2x + 0= 8

2x = 8

Divide both sides by 2 to isolate :

$2x \div 2 = 8 \div 2$

x = 4

The point of intersection is at (4,0) .

Similarly, The point of intersection of the -axis and the graph of 2x + y = 8 - the -intercept of the latter - can be found by substituting 0 for :

2x + y = 8

2 (0) + y = 8

0 + y= 8

y = 8

The point of intersection is at (0,8) .

The three vertices of the triangle are at the origin, (4,0) , and (0,8) . When this triangle is rotated about the -axis, the resulting solid of revolution is a cone whose base has radius r= 4 , and which has height h= 8 . Substitute these values into the formula for the volume of a cone:

$V = \frac{1}{3} \pi r^{2}h$

$= \frac{1}{3} \pi (4)^{2}(8)$

$= \frac{1}{3}\cdot 16 \cdot 8 \cdot \pi$

$= \frac{128}{3} \pi$

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Example Question #181 : Hi Set: High School Equivalency Test: Math

A right cone has slant height 20; its base has radius 10. Which of the following gives its volume to the nearest whole number?

(Round to the nearest whole number).

Possible Answers:

3,142

942

1,885

1,814

2,094

Correct answer:

1,814

Explanation:

The volume of a cone, given radius and height , can be calculated using the formula

$V = \frac{1}{3} \pi r^{2}h$ .

We are given that r = 10 , but we are not given the value of . We are given slant height l = 20 , and since the cone is a right cone, its radius, height, and slant height can be related using the Pythagorean relation

$r^{2}+ h^{2}= l^{2}$ .

Substituting 10 for and 20 for , we can find :

$10^{2}+ h^{2}= 20^{2}$

$100+ h^{2}=400$

$100+ h^{2} - 100 =400 - 100$

$h^{2} =300$

$h = \sqrt{300 }$

Now substitute in the volume formula:

$V = \frac{1}{3} \pi r^{2}h$

$= \frac{1}{3} \pi (10^{2}) \sqrt{300}$

$\approx \frac{1}{3} \cdot 3.1416 \cdot 100 \cdot 17.3$

$\approx 1813.8$

To the nearest whole, this is 1,814.

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Example Question #13 : Cones

The volume of a cone is $12.56\;\textup{cm}^3$ . Which of the following most closely approximates the radius of the base of the cone if its height is $1\;\textup{cm}$ ?

Possible Answers:

$\frac{3}{2}\;\textup{cm}$

$\sqrt{2}\;\textup{cm}$

$\sqrt{12}\;\textup{cm}$

$3\;\textup{cm}$

$\frac{3}{2}\pi\;\textup{cm}$

Correct answer:

$\sqrt{12}\;\textup{cm}$

Explanation:

The volume of a cone is given by the formula

$V= \frac{1}{3}\pi r^2h$

We are given that h=1 . Thus, the volume of our cone is given by

$V= \frac{1}{3}\pi r^2$ .

We are given that the volume of our cone is 12.56 .

Thus,

$12.56=\frac{1}{3}\pi r^2$

$\frac{12.56 \times3}{\pi} = r^2$

and

12 = r^2 .

Thus,

$\sqrt{12}=r$ .

so the correct answer is $\sqrt{12}\;cm$ .

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Example Question #1 : Spheres

The surface area of a sphere is equal to $100 \pi$ . Give the volume of the sphere.

Possible Answers:

$\frac{500}{3}\pi$

$\frac{1,000}{3}\pi$

$\frac{250 }{3}\pi$

$250 \pi$

$500 \pi$

Correct answer:

$\frac{500}{3}\pi$

Explanation:

The surface area of a sphere can be calculated using the formula

$A = 4 \pi r^{2}$

Solving for :

Set $A = 100 \pi$ and divide both sides by $4 \pi$ :

$4 \pi r^{2} = 100 \pi$

$\frac{4 \pi r^{2}}{4 \pi} = \frac{100 \pi}{4 \pi}$

$r^{2} =25$

Take the square root of both sides:

$r = \sqrt{25} = 5$

Set r=5 in the volume formula:

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

$V= \frac{4}{3} \pi \left (5 \right )^{3} = \frac{4}{3} \pi \left (125 \right ) = \frac{500}{3}\pi$ ,

the correct response.

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HiSET: Math : Use volume formulas to solve problems

Study concepts, example questions & explanations for HiSET: Math

All HiSET: Math Resources

Example Questions

Example Question #1 : Cones

Example Question #3 : Cones

Example Question #4 : Cones

Example Question #5 : Cones

Example Question #2 : Cones

Example Question #7 : Cones

Example Question #11 : Cones

Example Question #181 : Hi Set: High School Equivalency Test: Math

Example Question #13 : Cones

Example Question #1 : Spheres

All HiSET: Math Resources

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