Question

A baseball is covered with 42.3 cm2 of leather. Another larger baseball is to be constructed so that the ratio of the volume of the larger baseball to that of the smaller baseball is 16 over 3. What will be the surface area of the new baseball? Round your answer to the nearest tenth

Answer 1

We are told that the volumes between two spheres is 16/3, this means that:

$\frac{V_1}{V_2}=\frac{16}{3}$

Now the formula for the volume of a sphere is:

$V=\frac{4\pi r^3}{3}$

Replacing in the previous formula:

$\frac{\frac{4\pi r^3_1}{3}}{\frac{4\pi r^2_2}{3}}=\frac{16}{3}$

Simplifying:

$\frac{r^3_1}{r^3_2}=\frac{16}{3}$

Now we solve for the first radius which is the radius of the bigger sphere:

$r^3_1=\frac{16}{3}r^3_2$

Now we determine the radius of the smallest ball using the formula for the surface area of a sphere:

$A_S=4\pi r^2_2$

Now we replace the value of the surface area:

$42.3=4\pi r^2_2$

Now we solve for the radius:

$\frac{42.3}{4\pi}=r^2_2$

Taking square root on both sides:

$\sqrt[]{\frac{42.3}{4\pi}}=r_2$

Solving the operation:

$1.83\operatorname{cm}=r_2$

Now we replace the value of the radius in the proportion:

$r^3_1=\frac{16}{3}(1.83\operatorname{cm})^3$

Solving the operations:

$r^3_1=32.7\operatorname{cm}^3$

Taking cubic root to both sides:

$r_1=\sqrt[3]{32.7\operatorname{cm}^3}$

Solving the operation:

$r_1=3.19\operatorname{cm}$

Now we use the formula for the surface area using the new radius:

$A_S=4\pi r^2_1$

Replacing the radius:

$A_S=4\pi(3.19cm)^2$

Solving the operation:

$A_S=127.9\operatorname{cm}^2$

Therefore, the surface area of the new ball is 127.9 square centimeters.