Question

Find the rate of change of the surface area of a sphere with respect to the radius.

Answer 1

The rate of change of the surface area of a sphere with respect to the radius is
`\(4πr^2\),` where
`\(r\)` is the radius.

Step-by-step explanation:

The surface area
`(\(A\))` of a sphere is given by the formula
`\(A = 4πr^2\),`where
`\(r\)` is the radius. To find the rate of change of \(A\) with respect to
`\(r\),` we differentiate
`\(A\)` with respect to
`\(r\)`:

`\[ (dA)/(dr) = (d)/(dr)(4πr^2) \]`

Using the power rule of differentiation, where
`\( (d)/(dr)(x^n) = nx^(n-1) \), we get:`

`\[ (dA)/(dr) = 8πr \]`

Therefore, the rate of change of the surface area of a sphere with respect to the radius is
`\(8πr\).`However, if you're looking for the instantaneous rate of change, you can substitute a specific radius value into the expression \(8πr\).
`\(8πr\).` to get the exact rate at that point.

In practical terms, this means that as the radius of a sphere increases, the surface area increases at a rate proportional to twice the radius. Understanding such rates of change is crucial in various applications, from physics to engineering, where the relationship between geometric properties of objects and their behavior is essential for analysis and design.