Question

consider a ball with a radius of approximately 3 inches and a second ball with a radius of approximately 9 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? (round to the nearest tenth as needed.) the volume of the larger ball is 3053.6 in.³ (round to the nearest tenth as needed.) the surface - area - to - volume ratio for the smaller ball, 1.00, is smaller than the surface - area - to - volume ratio for the larger ball, . (round to the nearest hundredth as needed.)

Explanation:

Step1: Recall surface - area and volume formulas for a sphere

The surface - area formula of a sphere is $A = 4\pi r^{2}$, and the volume formula is $V=\frac{4}{3}\pi r^{3}$. The surface - area - to - volume ratio is $\frac{A}{V}=\frac{4\pi r^{2}}{\frac{4}{3}\pi r^{3}}=\frac{3}{r}$.

Step2: Calculate ratio for the smaller ball

For the smaller ball with $r_1 = 3$ inches, the surface - area - to - volume ratio $\frac{A_1}{V_1}=\frac{3}{r_1}=\frac{3}{3}=1.00$.

Step3: Calculate ratio for the larger ball

For the larger ball with $r_2 = 9$ inches, the surface - area - to - volume ratio $\frac{A_2}{V_2}=\frac{3}{r_2}=\frac{3}{9}\approx0.33$.

Step4: Compare the ratios

Since $1.00>0.33$, the surface - area - to - volume ratio for the smaller ball is larger.

Answer:

The surface - area - to - volume ratio for the smaller ball is 1.00. The surface - area - to - volume ratio for the larger ball is approximately 0.33. The surface - area - to - volume ratio for the smaller ball is larger.