Solid Geometry - ACT Math

Card 0 of 1422

Question

The radius of a cylinder is five and its height is nine. What is its volume?

Answer

To solve this question, you must remember that the formula for volume is the product of the area of the base and the height. The area of the base of this cylinder is $\pi r^2$ .

$V=A_b*h=\pi r^2h$

Plug in the given radius and height to solve.

$V=\pi(5)^2(9)$

$V=\pi(25)(9)$

$V=225\pi$

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Question

How much more volume can a cylinder hold than a cone given that both have the same radius and height?

Here B represents the area of the Base, and h the height.

Answer

Cylinder: Cone: $V=\frac{1}{3}Bh$

Thus the difference is 2/3Bh and that means a cylinder can hold 2/3Bh more given the same radius and height.

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Question

What is the volume of a round metal washer with an outer radius of 8 in, an inner radius of 2 in, and a thickness of 0.5 in?

Answer

The volume of a cylinder is given by the formula: $V=\pi r^2h$ .

For a shape with a hole through the center, the final volume is equal to the total volume of the shape minus the volume of the inner hole. In this question, we are looking for the volume given by the larger radius minus the volume given by the smaller radius. The height is equal to the thickness of the washer.

$V=\pi r_1^2h-\pi r_2^2h$

$V=\pi(8)^2(0.5)-\pi(2)^2(0.5)$

$V=\pi(64)(0.5)-\pi(4)(0.5)$

$V=32\pi-2\pi$

$V=30\pi\ in^3$

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Question

A certain cylinder has diameter that is twice the length of its height. If the volume of the cylinder is $64\pi$ cubic inches, what is its radius?

Answer

The volume of a cylinder is:

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

You can think of the volume as the area of the base times the height. Since it is given that the diameter is twice the length of the height, the radius (half the diameter) equals the height. If it helps to visualize these dimensions, draw the cylinder described.

The equation can be rewritten, using the height in terms of the radius.

$V=\pi r^2(r)$

$V=\pi r^{3}$

Plug in the given volume to solve for the radius.

$64\pi=\pi r^{3}$

$64=r^{3}$

$4=r$

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Question

The height of a right circular cylinder is and its radius is . What is the volume, in cubic meters, of the cylinder?

Answer

The volume of a right circular cylinder is equal to its height () multiplied by the area of the circle base ( $\pi r^2$ ).

In this scenario, the Volume

$V=5* (\pi *3^2) = 45\pi$ .

Therefore, the volume is $45\pi$ .

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Question

What is the volume of a cylinder with a radius of four inches and a height of seven inches?

Answer

Plug the radius and height into the formula for the volume of a cylinder:
$\V = \pi r^2h \newline V = \pi * 4^2 * 7 \newline V = 112 \pi inches^3$

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Question

What is the volume of a cylinder with a base diameter of 12 and a height of 3? Leave your answer in terms of $\pi$

Answer

To find the volume of a cylinder use formula:

$V = \pi r^2h$

For a cylinder with a radius of 6 and a height of 3 this yields:

$V = \pi 6^23$

$=108\pi$

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Question

A cylindrical tank is used as part of a water purifying plant. When contaminated water flows into the top section of the tank, pressure forces it through a mesh filter at the bottom of the tank and clean water exits through a funnel, leaving sediment behind. The tank's filter must be replaced when the total sediment content of the tank exceeds ten percent of the tank's total volume. If the tank is 100 feet tall and 18 feet in diameter, how much sediment, in cubic feet, can the drum hold before the filter must be changed?

Answer

The volume of a cylinder is found using the following formula:

$V = \pi r^2 h$

In this formula, the variable is the height of the cylinder and is its radius. Since the diameter is two times the radius, first solve for the radius.

$\text{Diameter}=2r$

Divide both sides of the equation by 2.

$\frac{18}{2}=\frac{2r}{}2$

The given cylinder has a radius of 9 feet. Now, substitute the calculated and known values into the equation for the volume of a cylinder and solve.

$V=\pi\times9^{2}\times 100$

$V=8100\pi$

This is the total volume of the tank. The question asks for the volume of ten percent of the tank—the point at which the filter must be replaced. To find this, move the decimal point in the numerical measure of total volume to the left one place in order to calculate ten percent of the total volume. (Ignore $\pi$ —you can treat it like a multiplier here. Since it appears on both sides of the equals sign, it doesn't affect the decimal shift.)

$8100\pi\times 0.10=810\pi$

The tank can hold $810\pi$ cubic feet of sediment before the filter needs to be changed.

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Question

Find the volume of a cylinder whose diameter is and height is .

Answer

To find volume, simply use the following formula. Remember, you were given diameter so radius is half of that. Thus,

$V=\pi{r^2}h=\pi3^28=72\pi$

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Question

Find the volume of cylinder with diameter of and height of .

Answer

To find volume of a cylinder, simply use the following formula. Thus,

$\textup{volume}=\pi{r^2}h=\pi(3^2)2=\pi92=18\pi$

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Question

Find the volume fo a cylinder whose radius is and height is .

Answer

To solve, simply use the formula. Thus,

$V=\pi{r^2}h=\pi\cdot1^2\cdot11=\pi\cdot1\cdot11=11\pi$

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Question

Find the volume of a cylinder with height 1 and radius 1.

Answer

To solve, simply use the formula for volume of a cylinder.

First, identify what is known.

Height = 1

Radius = 1

Substitute these values into the formula and solve.

Thus,

$V=\pi{r^2}h=\pi1^21=\pi$

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Question

Find the volume of a cylinder given height of and radius of .

Answer

To solve, simply use the following formula. Thus,

$V=\pi{r^2}h=\pi2^28=\pi48=32\pi$

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Question

A rectangular box has two sides with the following lengths:

and

If it possesses a volume of $84$ cm$^{3}$ , what is the area of its largest side?

Answer

The volume of a rectangular prism is found using the following formula:

$V=l\times w\times h$

If we substitute our known values, then we can solve for the missing side.

$84=3\times 4\times x$

Divide both sides of the equation by 12.

$\frac{84}{12}=\frac{12x}{12}$

We now know that the missing length equals 7 centimeters.

This means that the box can have sides with the following dimensions: 3cm by 4cm; 7cm by 3cm; or 7cm by 4cm. The greatest area of one side belongs to the one that is 7cm by 4cm.

$A=l\times w$

$A=4\times 7$

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Question

A box's length is twice as long as its width. Its height is the sum of its length and its width. What is the volume of this box if its length is 10 units?

Answer

The formula for the volume of a rectangular prism is , where "" is volume, "" is length, "" is width and "" is height.

We know that and . By rearranging , we get $W=\frac{L}{2}$ . Substituting $\frac{L}{2}$ into the volume equation for and into the same equation for , we get the following:

$V = L \cdot \left(\frac{L}{2}\right)\cdot (L+\left(\frac{L}{2}\right))$

$V = L\cdot \left(\frac{L}{2}\right)\cdot \frac{3L}{2}$

$V = 10 \cdot 5 \cdot 15$

units cubed

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Question

A rectangular prism has the following dimensions:

Length:

Width:

Height:

Find the volume.

Answer

Given that the dimensions are: , , and and that the volume of a rectangular prism can be given by the equation:

$v=l \cdot w \cdot h$ , where is length, is width, and is height, the volume can be simply solved for by substituting in the values.

This final value can be approximated to .

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Question

Solve for the volume of a prism that is 4m by 3m by 8m.

Answer

The volume of the rectangle

so we plug in our values and obtain

.

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Question

Find the volume of a regular tetrahedron if one of its edges is $\sqrt[3]{6}:cm$ long.

Answer

Write the volume equation for a tetrahedron.

$V=\frac{e^3}{6\sqrt2}$

In this formula, stands for the tetrahedron's volume and stands for the length of one of its edges.

Substitute the given edge length and solve.

$V=\frac{(\sqrt[3]6:cm)^3}{6\sqrt2} = \frac{6:cm^3}{6\sqrt2}= \frac{1}{\sqrt2}:cm$

Rationalize the denominator.

$\frac{1}{\sqrt2}:cm\cdot \frac{\sqrt2}{\sqrt2} = \frac{\sqrt2}{2}:cm$

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Question

Find the volume of a tetrahedron if the side length is $\frac{1}{6}$ .

Answer

Write the equation to find the volume of a tetrahedron.

$V=\frac{a^3}{6\sqrt2}$

Substitute the side length and solve for the volume.

$V=\frac{(\frac{1}{6})^3}{6\sqrt2}= \frac{1}{216}\left(\frac{1}{6\sqrt2}\right)= \frac{1}{1296\sqrt2}$

Rationalize the denominator.

$V=\frac{1}{1296\sqrt2} \cdot \frac{\sqrt2}{\sqrt2}= \frac{\sqrt2}{1296\cdot 2} = \frac{\sqrt2}{2592}$

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Question

What is the volume of a regular tetrahedron with an edge length of 6?

Answer

The volume of a tetrahedron can be solved for by using the equation:

$V= \frac{a^3}{6\sqrt{2}}$

where is the measurement of the edge of the tetrahedron.

This problem can be quickly solved by substituting 6 in for .

$V=\frac{6^3}{6\sqrt{2}}=\frac{216}{6\sqrt{2}}=\frac{36}{\sqrt{2}}$

${\color{Blue} V=25.5}$

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