Question

the volume of a sphere is a function of its radius, $v = \frac{4}{3}\pi r^3$. evaluate the function for the volume of a volleyball with radius 10.7 cm. the volume is \\(\square\\) \\(\mathrm{cm}^3\\). (round to the nearest tenth as needed.)

Explanation:

Step1: Substitute the radius value into the formula

Given the formula for the volume of a sphere \( V=\frac{4}{3}\pi r^{3} \), and \( r = 10.7\space cm \). Substitute \( r=10.7 \) into the formula:
\( V=\frac{4}{3}\pi(10.7)^{3} \)

Step2: Calculate \( (10.7)^{3} \)

First, calculate \( 10.7\times10.7\times10.7 \).
\( 10.7\times10.7 = 114.49 \), then \( 114.49\times10.7=114.49\times(10 + 0.7)=114.49\times10+114.49\times0.7 = 1144.9+80.143 = 1225.043 \)

Step3: Multiply by \( \frac{4}{3}\pi \)

Now, multiply \( 1225.043 \) by \( \frac{4}{3}\pi \).
\( \frac{4}{3}\times1225.043\times\pi=\frac{4900.172}{3}\times\pi\approx1633.3907\times\pi \)
Using \( \pi\approx3.1416 \), we get \( 1633.3907\times3.1416\approx1633.3907\times3 + 1633.3907\times0.1416\)
\( 1633.3907\times3 = 4900.1721 \)
\( 1633.3907\times0.1416\approx1633.3907\times0.1+1633.3907\times0.04 + 1633.3907\times0.0016=163.3391+65.3356+2.6134 = 231.2881 \)
Adding them together: \( 4900.1721+231.2881 = 5131.4602 \) (rounded to the nearest tenth)

Answer:

\( 5131.5 \) (Note: The slight difference may come from different precision in \( \pi \) value. Using a calculator directly: \( V=\frac{4}{3}\pi(10.7)^3\approx\frac{4}{3}\times3.14159265\times1225.043\approx5131.5 \) when rounded to the nearest tenth)