The area A = πr2 of a circular puddle changes with the radius. At what rate does the area change with respect to the radius when r = 5ft?
To find the rate at which the area changes with respect to the radius, we need to calculate the derivative of the area equation with respect to the radius. In this case, the area of a circle is given by the equation A = πr^2.
To find the derivative, we can use the power rule of differentiation. According to the power rule, if we have a function in the form f(x) = ax^n, where a and n are constants, the derivative of the function f(x) with respect to x is given by f'(x) = anx^(n-1).
In this case, the area function is A = πr^2, where π is a constant and r is the radius. Applying the power rule, we can find the derivative of A with respect to r (denoted as dA/dr):
dA/dr = d(πr^2)/dr
Using the power rule, we can rewrite this as:
dA/dr = 2πr^(2-1)
Simplifying further, we have:
dA/dr = 2πr
Now, we can substitute r = 5ft into this derivative equation to find the rate at which the area changes with respect to the radius when r = 5ft:
dA/dr = 2π(5)
dA/dr = 10π
Therefore, when the radius is 5ft, the rate at which the area changes with respect to the radius is 10π square feet per foot.