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Word Problems: Volume and Surface Area of Spheres
In order to solve problems which require application of the volume and surface area for spheres, it is necessary to
  • know how to simplify and solve expressions involving squares and cubes
A typical problem involving the volume or surface area of a sphere gives us various information about the size of the sphere - usually one or more values for the volume, surface area, radius, or diameter. You will then be asked to calculate the other missing values based on this given information.
Suppose the diameter of a sphere is 10 inches. What are its volume and surface area?
To get started, notice that the formulas for volume and surface area involve the radius and not the diameter. The radius is half the diameter or 5 inches.
We use r = 5 in the formulas for volume and surface area.
Volume: V = cubic inches
Surface area: S = square inches
It is difficult to inspect these answers for reasonableness. Care must be taken in arithmetic when squaring and cubing the radius.
Notice also that both formulas involve use of 4p. If we rewrite the formula for volume, we can get a relationship between volume and surface area:
where S is the surface area. This is not often a helpful relationship, but we may refer to it when comparing volume and surface area.
For example, suppose the surface area of a sphere equals twice the numerical value of its volume. What is the sphere's radius?

r = 3/2 = 1.5

Let's Practice

Question #1
AudioSuppose the surface area of a sphere is 64p square feet. Find the radius and volume of this sphere.

Question #2
AudioThe volume of a sphere is 288p cubic centimeters. What are the radius and surface area?

Try These
Question #1
AudioA sphere has a diameter of 8 feet. What are its volume and surface area?

Question #2
AudioThe surface area of a sphere is 28p. What are the radius and the volume of this sphere?

The formulas for surface area and volume involve use of the radius. It is important to be using the radius and not the diameter of the sphere. Finding the radius if we are given volume or surface area involves taking a square or cube root. The biggest hurdle with this type of problem is doing the arithmetic carefully. Note that the surface area formula uses r squared (r2) and the volume formula uses r cubed (r3). Correspondingly, remember that surface area is measured in square units and volume is measured in cubic units.
M Ransom

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