Exercise 43 Page 854

Use the formula for the surface area of a sphere and the formula for the volume of a sphere.

The surface area is 9 times smaller and the volume is 27 times smaller.

Practice makes perfect

Let r be the radius of a sphere.

We are asked to describe how dividing the radius of the sphere by 3 affects the surface area and the volume of the sphere. Therefore, the radius of the new sphere is r_\text{new}=\frac{1}{3}r.

Let's use the formula for the surface area of a sphere and for the volume of a sphere.

Sphere	Old	New
Radius	r	1/3r
Area	A=4π Radius^2
Area	\textcolor{darkorange}{A_\text{old}}=\textcolor{darkorange}{4\pi r^2}	A_\text{new}=4\pi \left({\color{#0000FF}{\dfrac{1}{3}r}}\right)^2=\dfrac{1}{9}\cdot \textcolor{darkorange}{4\pi r^2}
Volume	V=4/3π Radius^3
Volume	\textcolor{darkviolet}{V_\text{old}}=\textcolor{darkviolet}{\dfrac{4}{3}\pi r^3}	V_\text{new}=\dfrac{4}{3}\pi \left({\color{#0000FF}{\dfrac{1}{3}r}}\right)^3=\dfrac{1}{27}\cdot\textcolor{darkviolet}{\dfrac{4}{3}\pi r^3}

From the table above we get that A_\text{new}=\frac{1}{9}\textcolor{darkorange}{A_\text{old}} and V_\text{new}=\frac{1}{27}\textcolor{darkviolet}{V_\text{old}}. This tells us that the surface area of the sphere is 9 times smaller and the volume of the sphere is 27 times smaller if we divide the radius by 3.

Subchapter links

Exercises p.852-855

Exercise 43 Page 854

Hints

Check the answer

Practice exercises

Assess your skill