Cones - HiSet: High School Equivalency Test: Math

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Question

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 $1;\textup{cm}$ ?

Answer

The volume of a cone is given by the formula

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

We are given that . 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 .

Thus,

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

so

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

and

.

Thus,

$\sqrt{12}=r$ .

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

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Question

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

.

Give the volume of the solid of revolution formed.

Answer

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

The point of intersection of the -axis and the graph of - the -intercept of the latter - can be found by substituting 0 for :

Divide both sides by 2 to isolate :

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

The point of intersection is at .

Similarly, The point of intersection of the -axis and the graph of - the -intercept of the latter - can be found by substituting 0 for :

The point of intersection is at .

The three vertices of the triangle are at the origin, , and . When this triangle is rotated about the -axis, the resulting solid of revolution is a cone whose base has radius , and which has height . 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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Question

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

.

Give the volume of the solid of revolution formed.

Answer

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

The point of intersection of the -axis and the graph of —the -intercept of the latter—can be found by substituting 0 for :

Divide both sides by 2 to isolate :

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

The point of intersection is at .

Similarly, the point of intersection of the -axis and the graph of —the -intercept of the latter—can be found by substituting 0 for :

The point of intersection is at .

The three vertices of the triangle are at the origin, , and . When this triangle is rotated about the -axis, the resulting solid of revolution is a cone whose base has radius , and which has height . 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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Question

About the -axis, rotate the triangle with its vertices at , , and the origin. What is the volume of the solid of revolution formed?

Answer

When this triangle is rotated about the -axis, the resulting solid of revolution is a cone whose base has radius , and which has height . Substitute these values into the formula for the volume of a cone:

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

$= \frac{1}{3} \pi (5^{2}) (6)$

$= 25 \cdot 6 \cdot \frac{1}{3} \pi$

$= 50 \pi$

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Question

About the x-axis, rotate the triangle with its vertices at , , and the origin. What is the volume of the solid of revolution formed?

Answer

When this triangle is rotated about the -axis, the resulting solid of revolution is a cone whose base has radius , and which has height . Substitute these values into the formula for the volume of a cone:

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

$= \frac{1}{3} \pi (6)^{2}(5)$

$= 36 \cdot 5 \cdot \frac{1}{3} \pi$

$=60 \pi$

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Question

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).

Answer

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 , but we are not given the value of . We are given slant height , 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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Question

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).

Answer

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 and , 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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Question

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

Answer

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 , but we are not given the value of . We are given that , 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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Question

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.

Answer

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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Question

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?

Answer

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 .

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

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

Set and :

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

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

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

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

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

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

.

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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Question

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?

Answer

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 .

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

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

Set and :

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

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

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

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

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

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

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Question

A right square pyramid has height 10 and a base of perimeter 36.

Inscribe a right cone inside this pyramid. What is its volume?

Answer

The length of one side of the square is one fourth of its perimeter, or $36 \div 4 = 9$ . The cone inscribed inside this pyramid has the same height. Its base is the circle inscribed inside the square. This circle will have as its diameter the length of one side of the square, or 9, and, as its radius, half this, or $9 \div 2 = \frac{9}{2}$ .

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

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

Set $r = \frac{9}{2}$ and :

$V = \frac{1}{3} \pi \left ( \frac{9}{2} \right )^{2} 10$

$= \frac{1}{3} \left ( \frac{81}{4} \right ) 10 \pi$

$= \frac{135}{2} \pi$

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Question

A right square pyramid has height 10 and a base of area 36.

Inscribe a right cone inside this pyramid. What is its volume?

Answer

The length of one side of the square is the square root of the area, or $\sqrt{36} = 6$ . The cone inscribed inside this pyramid will have as its base the circle inscribed inside the square. This circle will have as its diameter the length of one side of the square, or 6, and, as its radius, half this, or 3.

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

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

Set and :

$V = \frac{1}{3} \pi \left (3 \right )^{2} 10$

$= \frac{1}{3} \left ( 9 \right ) 10 \pi$

$=30 \pi$

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