Question

A spherical balloon with a radius rinches has volume\(V\left( r \right) = \frac{4}{3}\pi {r^3}\). Find a function that represents the amount of air required to inflate the balloon from a radius of rinches to a radius of \(r + 1\)inches.

Short answer

The required function is\(F\left( r \right) = \frac{4}{3}\pi \left( {3{r^2} + 3r + 1} \right)\).

Step-by-step solution

Describe the given information

The volume is given as,

\(V\left( r \right) = \frac{4}{3}\pi {r^3}\)

The change in radius is from r to\(r + 1\).

Find a function that represents the amount of air required to inflate the balloon

Let \(F\left( r \right)\) denote the amount of air required to inflate the balloon from a radius of \(r\)to\(r + 1\).

The \(F\left( r \right)\) is given by,

\(\begin{array}{c}F\left( r \right) = V\left( {r + 1} \right) - V\left( r \right)\\ = \frac{4}{3}\pi {\left( {r + 1} \right)^3} - \frac{4}{3}\pi {r^3}\\ = \frac{4}{3}\pi \left( {{{\left( {r + 1} \right)}^3} - {r^3}} \right)\\ = \frac{4}{3}\pi \left( {3{r^2} + 3r + 1} \right)\end{array}\)

Therefore, the required function is\(F\left( r \right) = \frac{4}{3}\pi \left( {3{r^2} + 3r + 1} \right)\).