HSPT Math : Geometry
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Example Questions
Example Question #1 : Volume Of A Three Dimensional Figure
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)?
The volume of a sphere, given its radius, is
Set 
The diameter is twice this, or 12.4 yards.
Example Question #1 : How To Find The Volume Of A Sphere
The diameter of a sphere is 
The diameter of a sphere is 
The volume enclosed by a sphere is given by the formula:
Example Question #2 : How To Find The Volume Of A Sphere
A spherical balloon has a diameter of 10 meters. Give the volume of the balloon.
The volume enclosed by a sphere is given by the formula:
where 
Example Question #4 : Volume Of A Three Dimensional Figure
The volume of a sphere is 1000 cubic inches. What is the diameter of the sphere.
The volume of a sphere is:
Where 
So we can get:
Example Question #1 : Solid Geometry
Give the volume of the above box in cubic centimeters.
100 centimeters make one meter, so convert each of the dimensions of the box by multiplying by 100.
Use the volume formula, substituting 
Example Question #2 : Solid Geometry
A cube made of nickel has sidelength 20 centimeters, Nickel has a density of 8.9 grams per cubic centimeter. What is the mass of this cube?
The volume of the cube is 
Example Question #1 : Volume Of A Rectangular Solid
A rectangular prism has length 24 inches, width 18 inches, and height 15 inches. Give its volume in cubic feet.
Divide each dimension in inches by 12 to convert from inches to feet:
Multiply the three to get the volume:
Example Question #3 : Solid Geometry
Give the volume of the box in cubic inches.
Use the volume formula, substituting 
Example Question #1 : How To Find The Volume Of A Sphere
A cube with sides of 4” each contains a floating sphere with a radius of 1”. What is the volume of the space outside of the sphere, within the cube?
11.813 in3
4.187 in3
64 in3
59.813 in3
11.813 in3
59.813 in3
Volume of Cube = side3 = (4”)3 = 64 in3
Volume of Sphere = (4/3) * π * r3 = (4/3) * π * 13 = (4/3) * π * 13 = (4/3) * π = 4.187 in3
Difference = Volume of Cube – Volume of Sphere = 64 – 4.187 = 59.813 in3
Example Question #2 : How To Find The Volume Of A Sphere
If a sphere's diameter is doubled, by what factor is its volume increased?
3
4
8
2
16
8
The formula for the volume of a sphere is 4πr3/3. Because this formula is in terms of the radius, it would be easier for us to determine how the change in the radius affects the volume. Since we are told the diameter is doubled, we need to first determine how the change in the diameter affects the change in radius.
Let us call the sphere's original diameter d and its original radius r. We know that d = 2r.
Let's call the final diameter of the sphere D. Because the diameter is doubled, we know that D = 2d. We can substitute the value of d and obtain D = 2(2r) = 4r.
Let's call the final radius of the sphere R. We know that D = 2R, so we can now substituate this into the previous equation and write 2R = 4r. If we simplify this, we see that R = 2r. This means that the final radius is twice as large as the initial radius.
The initial volume of the sphere is 4πr3/3. The final volume of the sphere is 4πR3/3. Because R = 2r, we can substitute the value of R to obtain 4π(2r)3/3. When we simplify this, we get 4π(8r3)/3 = 32πr3/3.
In order to determine the factor by which the volume has increased, we need to find the ratio of the final volume to the initial volume.
32πr3/3 divided by 4πr3/3 = 8
The volume has increased by a factor of 8.
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