the volume of a sphere is a function of its radius, $v = \\frac{4}{3}\\pi r^3$. evaluate the function for…

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)