All SAT II Math II Resources
Example Questions
Example Question #1 : 3 Dimensional Geometry
The width of a box is two-thirds its height and three-fifths its length. The volume of the box is 6 cubic meters. To the nearest centimeter, give the width of the box.
Call , , and the length, width, and height of the crate.
The width is two-thirds the height, so
.
Equivalently,
The width is three-fifths the length, so
.
Equivalently,
The dimensions of the crate in terms of are , , and . The volume is their product:
,
Substitute:
Taking the cube root of both sides:
meters.
Since one meter comprises 100 centimeters, multiply by 100 to convert to centimeters:
centimeters,
which rounds to 134 centimeters.
Example Question #1 : Volume
The shaded face of the rectangular prism in the above diagram is a square. The volume of the prism is ; give the value of in terms of .
The volume of a rectangular prism is the product of its length, its width, and its height; that is,
Since the shaded face of the prism is a square, we can set , and ; substituting and solving for :
Taking the positive square root of both sides, and simplifying the expression on the right using the Quotient of Radicals Rule:
Example Question #11 : 3 Dimensional Geometry
Find the volume of a sphere with a diameter of 10.
The surface area of a sphere is found using the formula . We are given the diameter of the circle and so we have to use it to find the radius (r).
Plug r into the formula to find the surface area
Example Question #11 : Volume
Determine the volume of the cube with a side length of .
Write the formula for the volume of a cube.
Substitute the length into the formula.
The volume is:
Example Question #11 : Volume
Billy has a ice cream cone that consists of a cone and hemisphere. Suppose the cone has a height of 4 inches, and the radius of the hemisphere is 2 inches. Assuming that the combined shape is not irregular, what is the total volume?
Write the volume for a cone.
Substitute the radius and height. The radius is 2.
Write the volume for a hemisphere. This should be half the volume of the full sphere.
Substitute the radius.
Add the volumes of the cone and hemisphere to determine the total volume.
The answer is:
Example Question #391 : Sat Subject Test In Math Ii
Find the volume of a sphere with a diameter of .
Divide the diameter by two to get the radius. This is also the same as multiplying the diameter by one-half.
Write the formula for the volume of the sphere.
Substitute the radius.
Simplify the terms.
The answer is:
Example Question #13 : 3 Dimensional Geometry
If the side of a cube is , what must be the volume?
Write the formula for the volume of a cube.
Substitute the side length. When we are multiplying common bases with exponents, we are adding the exponents instead.
The answer is:
Example Question #14 : 3 Dimensional Geometry
Determine the volume of a cube if the side length is .
Write the formula for the volume of a cube.
Substitute the side length into the equation.
The answer is:
Example Question #15 : 3 Dimensional Geometry
The radius and the height of a cylinder are equal. If the volume of the cylinder is , what is the diameter of the cylinder?
Recall how to find the volume of a cylinder:
Since we know that the radius and the height are equal, we can rewrite the equation:
Using the given volume, find the length of the radius.
Since the question asks you to find the diameter, multiply the radius by two.
Example Question #11 : Volume
Determine the volume of the cube if the side lengths are .
The volume of a cube is:
Substitute the dimensions.
The answer is: