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HiSET: Math : Cones

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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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HiSET: Math : Cones

Study concepts, example questions & explanations for HiSET: Math

All HiSET: Math Resources

Example Questions

Example Question #11 : Cones

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

Example Question #13 : Cones

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