Exercise 37 Page 653

Radius	Surface Area	Volume
3 in.	36 π in.^2	36 π in.^3
6 in.	144π in.^2	288 π in.^3
9 in.	324π in.^2	972 π in.^3
12 in.	576π in.^2	2304 π in.^3

Practice makes perfect

For spheres with different radii, we want to complete a table with the measures of their surface areas and volumes.

Radius	Surface area	Volume
3 in.	36 π in.^2	36 π in.^3
6 in.
9 in.
12 in.

The surface area of a sphere is four times the product of π and the radius squared.

S=4π r^2 To find the surface area of a sphere with a radius of 6 inches we will substitute r for 6 in this formula. S=4π r^2 ⇒ S=4π ( 6)^2=144π We find the surface area of the sphere with a radius of 9 inches and the one with a radius of 12 inches in the same way.

Radius	S=4π r^2	Surface area
3	4π ( 3)^2	36 π
6	4π ( 6)^2	144π
9	4π ( 9)^2	324π
12	4π ( 12)^2	576π

The formula for the volume of a sphere is four thirds times the product of π and the radius cubed. V=4/3π r^3 Let's use this formula to find the volume of a sphere with a radius of 6 inches. V=4/3π r^3 ⇒ V=4/3π ( 6)^3=288π We will use the same formula to find the volume of a sphere with radius 9 inches and that of a sphere with a radius of 12 inches.

Radius	V=4/3π r^3	Volume
3	4/3π ( 3)^3	36 π
6	4/3π ( 6)^3	288π
9	4/3π ( 9)^3	972π
12	4/3π ( 12)^3	2304π

With the results we have gathered we can complete the given table.

Radius	Surface area	Volume
3 in.	36 π in.^2	36 π in.^3
6 in.	144π in.^2	288 π in.^3
9 in.	324π in.^2	972 π in.^3
12 in.	576π in.^2	2304 π in.^3

To study what happens to the surface area for various radii we can split the results from Part A into factors.

Radius	Radius factored	Surface area	Surface area factored
3	1* 3	36 π	1* 36 π
6	2* 3	144π	4* 36 π
9	3* 3	324π	9* 36 π
12	4* 3	576π	16* 36 π

In the table we can see that when the radius is doubled the surface area is grows by a factor of 4. For a tripled radius, the surface area is nine times larger. Finally, a quadrupled radius results in a sphere with a 16 times larger surface area.

Next we want to know what happens to the volume of a sphere for various radii. Again, we can study this by splitting the results from Part A into factors.

Radius	Radius factored	Volume	Volume factored
3	1* 3	36 π	1* 36 π
6	2* 3	288π	8* 36 π
9	3* 3	972π	27* 36 π
12	4* 3	2304π	64* 36 π

We look in the table and see that when the radius is doubled the volume grows by a factor of 8. If the radius is tripled, the volume is 27 times larger. For a quadrupled radius, the volume is 64 times larger.

Subchapter links

Communicate Your Answer p.647

Monitoring Progress p.649-651

Exercise 37 Page 653

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