Question

consider a ball with a radius of approximately 5 inches and a second ball with a radius of approximately 20 inches. compute the approximate surface area and volume for both balls. then find the surface - area - to - volume ratio for both balls. which ball has the larger ratio? the surface area of the smaller ball is (round to the nearest tenth as needed.)

Explanation:

Step1: Recall surface - area formula for a sphere

The surface - area formula of a sphere is $A = 4\pi r^{2}$. For the smaller ball with $r_1=5$ inches, $A_1 = 4\pi r_1^{2}=4\pi\times5^{2}=4\pi\times25 = 100\pi$ square inches.

Step2: Recall volume formula for a sphere

The volume formula of a sphere is $V=\frac{4}{3}\pi r^{3}$. For the smaller ball with $r_1 = 5$ inches, $V_1=\frac{4}{3}\pi r_1^{3}=\frac{4}{3}\pi\times5^{3}=\frac{4}{3}\pi\times125=\frac{500}{3}\pi$ cubic inches.

Step3: Calculate surface - area - to - volume ratio for the smaller ball

The surface - area - to - volume ratio $R_1=\frac{A_1}{V_1}=\frac{100\pi}{\frac{500}{3}\pi}$. The $\pi$ terms cancel out, and $R_1=\frac{100}{\frac{500}{3}}=\frac{100\times3}{500}=\frac{3}{5}=0.6$.

For the larger ball with $r_2 = 20$ inches:

Step4: Calculate surface area of the larger ball

$A_2=4\pi r_2^{2}=4\pi\times20^{2}=4\pi\times400 = 1600\pi$ square inches.

Step5: Calculate volume of the larger ball

$V_2=\frac{4}{3}\pi r_2^{3}=\frac{4}{3}\pi\times20^{3}=\frac{4}{3}\pi\times8000=\frac{32000}{3}\pi$ cubic inches.

Step6: Calculate surface - area - to - volume ratio for the larger ball

$R_2=\frac{A_2}{V_2}=\frac{1600\pi}{\frac{32000}{3}\pi}$. The $\pi$ terms cancel out, and $R_2=\frac{1600}{\frac{32000}{3}}=\frac{1600\times3}{32000}=\frac{4800}{32000}=0.15$.

Since $0.6>0.15$, the smaller ball has the larger ratio.

Answer:

The surface - area - to - volume ratio of the smaller ball is $0.6$. The smaller ball has the larger surface - area - to - volume ratio.