Final answer:
The rate at which the area of a circle changes with respect to the radius when r = 3ft is calculated by taking the derivative of the area function, resulting in an approximate value of 18.85 ft²/ft.
Step-by-step explanation:
To figure out at what rate the area changes with respect to the radius when r = 3ft, we need to use calculus, specifically the concept of a derivative. The area of a circle is given by the formula A = πr², and the rate of change of the area with respect to the radius is the derivative of the area with respect to the radius (dA/dr). So, let's calculate it.
dA/dr = d(πr²)/dr = 2πr
When r = 3ft, the rate at which the area changes with respect to the radius is:
dA/dr = 2π(3ft) = 6π ft²/ft
Since π is approximately 3.14159, this gives us:
dA/dr ≈ 6(3.14159)ft²/ft ≈ 18.84954 ft²/ft
Therefore, at r = 3ft, the area changes at a rate of approximately 18.85 ft²/ft.