Volume of a Three-Dimensional Figure - SSAT Upper Level Quantitative

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Question

What is the volume of a sphere with diameter 12 feet?

Answer

The radius of the sphere is half the diameter, or 6 feet; use the formula

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

Setting :

$V= \frac{4}{3} \pi \cdot 6^{3} \approx904.8 ft^{3}$

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Question

A car dealership wants to fill a large spherical advertising ballon with helium. It can afford to buy 1,000 cubic yards of helium to fill this balloon. What is the greatest possible diameter of that balloon (nearest tenth of a yard)?

Answer

The volume of a sphere, given its radius, is

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

Set , solve for , and double that to get the diameter.

$1,000= \frac{4\pi r^{3}}{3}$

$r^{3} = \frac{1,000 \cdot 3 }{4\pi} \approx 238.7$

$r \approx \sqrt[3]{238.7} \approx 6.2$

The diameter is twice this, or 12.4 yards.

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Question

A spherical balloon has a diameter of 10 meters. Give the volume of the balloon.

Answer

The volume enclosed by a sphere is given by the formula:

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

where is the radius of the sphere. The diameter of the balloon is 10 meters so the radius of the sphere would be $10\div 2=5$ meters. Now we can get:

$Volume=\frac{4}{3}\pi r^3=\frac{4}{3}\pi\times 5^3=\frac{4}{3}\times \pi \times 125\approx 523.6 m^3$

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Question

The volume of a sphere is 1000 cubic inches. What is the diameter of the sphere.

Answer

The volume of a sphere is:

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

Where is the radius of the sphere. We know the volume and can solve the formula for :

$Volume=\frac{4}{3}\pi r^3=1000\Rightarrow r^3=\frac{3\times1000 }{4\pi}\approx 238.85\Rightarrow r\approx 6.2$ inches

So we can get:

$Diameter=2r=2\times 6.2=12.4inches$

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Question

The diameter of a sphere is . Give the volume of the sphere in terms of .

Answer

The diameter of a sphere is so the radius of the sphere would be $6t\div 2=3t$

The volume enclosed by a sphere is given by the formula:

$Volume=\frac{4}{3}\pi r^3=\frac{4}{3}\pi \times (3t)^3=\frac{4}{3}\pi\times 27t^3\Rightarrow Volume=36\pi t^3$

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Question

A sphere has a diameter of inches. What is the volume of this sphere?

Answer

To find the volume of a sphere, use the following formula:

$\text{Volume}=\frac{4}{3}\pi\times r^3$ , where is the radius of the sphere.

Now, because we are given the diameter of the sphere, divide that value in half to find the radius.

$r=10\div2=5$

Now, plug this value into the volume equation.

$\text{Volume}=\frac{4}{3}\pi\times 5^3$

$\text{Volume}=\frac{4}{3}\pi\times 125$

$\text{Volume}=\frac{500}{3}\pi=523.6$

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Question

What is the volume of a sphere with a diameter of ?

Answer

Write the formula for the volume of the sphere.

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

The radius is half the diameter, which is five. Substitute the value.

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

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Question

Find the volume of a regular hexahedron with a side length of .

Answer

A regular hexahedron is another name for a cube.

To find the volume of a cube,

$\text{Volume}=(side)^3$

Plugging in the information given in the question gives

$\text{Volume}=(14)^3=2744$

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Question

Find the volume of a regular octahedron with side lengths of .

Answer

Use the following formula to find the volume of a regular octahedron:

$\text{Volume}=\frac{\sqrt2}{3}(side)^3$

Plugging in the information from the question,

$\text{Volume}=\frac{\sqrt2}{3}(9t)^3=243t^3\sqrt2$

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Question

Find the volume of a regular hexagonal prism that has a height of . The side length of the hexagon base is .

Answer

The formula to find the volume of a hexagonal prism is

$\text{Volume}=\frac{3\sqrt3}{2}(side)^2(height)$

Plugging in the values given by the question will give

$\text{Volume}=\frac{3\sqrt3}{2}(12^2)(10)=2160\sqrt3$

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Question

Find the volume of a prism that has a right triangle base with leg lengths of and and a height of .

Answer

To find the volume of a prism, multiply the area of the base by the height.

$\text{Area of Base}=\frac{3\times8}{2}=12$

$\text{Volume}=12\times height=12\times12=144$

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Question

Find the volume of a square pyramid that has a height of and a side length of .

Answer

The formula to find the volume of a square pyramid is

$\text{Volume}=(side)^2\frac{height}{3}$

So plugging in the information given from the question,

$\text{Volume}=4^2\frac{9}{3}=16\times3=48$

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Question

Find the volume of a square pyramid with a height of and a length of a side of its square base of .

Answer

The formula to find the volume of a square pyramid is

$\text{Volume}=(side)^2\frac{height}{3}$

So plugging in the information given from the question,

$\text{Volume}=(3^2)\frac{12}{3}=9\times 4=36$

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Question

Find the volume of a regular hexagonal prism that has a height of . The side length of the hexagon base is .

Answer

The formula to find the volume of a hexagonal prism is

$\text{Volume}=\frac{3\sqrt3}{2}(side)^2(height)$

Plugging in the values given by the question will give

$\text{Volume}=\frac{3\sqrt3}{2}(5^2)(4)=150\sqrt3$

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Question

In terms of , find the volume of a regular hexagonal prism that has a height of . The hexagon base has side lengths of .

Answer

The formula to find the volume of a hexagonal prism is

$\text{Volume}=\frac{3\sqrt3}{2}(side)^2(height)$

Plugging in the values given by the question will give

$\text{Volume}=\frac{3\sqrt3}{2}(4x)^2(5)=120x^2\sqrt3$

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Question

Find the volume of a regular tetrahedron that has side lengths of .

Answer

The formula to find the volume of a tetrahedron is

$\text{Volume}=\frac{\sqrt2}{12}(side)^3$

Plugging in the information given by the question gives

$\text{Volume}=\frac{\sqrt2}{12}(3a)^3=\frac{27a^3\sqrt2}{12}=\frac{9a^3\sqrt2}{4}$

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Question

Find the volume of a regular tetrahedron with a side length of .

Answer

The formula to find the volume of a tetrahedron is

$\text{Volume}=\frac{\sqrt2}{12}(side)^3$

Plugging in the information given by the question gives

$\text{Volume}=\frac{\sqrt2}{12}(12)^3=144\sqrt2$

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Question

Find the volume of a regular octahedron that has a side length of .

Answer

Use the following formula to find the volume of a regular octahedron:

$\text{Volume}=\frac{\sqrt2}{3}(side)^3$

Plugging in the information from the question,

$\text{Volume}=\frac{\sqrt2}{3}(8)^3=\frac{512\sqrt2}{3}$

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Question

Find the volume of a regular octahedron that has a side length of .

Answer

Use the following formula to find the volume of a regular octahedron:

$\text{Volume}=\frac{\sqrt2}{3}(side)^3$

Plugging in the information from the question,

$\text{Volume}=\frac{\sqrt2}{3}(6y)^3=72y^3\sqrt2$

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Question

A cube has six square faces, each with area 64 square inches. Using the conversion factor 1 inch = 2.5 centimeters, give the volume of this cube in cubic centimeters, rounding to the nearest whole number.

Answer

The volume of a cube is the cube of its sidelength, which is also the sidelength of each square face. This sidelength is the square root of the area 64:

$\sqrt{64}=8$ inches.

Multiply this by 2.5 to get the sidelength in centimeters:

$\times 2.5$ centimeters.

The cube of this is

$20^{3}$ cubic centimeters

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