The area A = πr2 of a circular water ripple changes with the radius. At what rate does the area change with respect to the radius when r = 4ft?
A = π r^2
dA/dr = 2πr
so when r = 4
dA/dr = 8π
Why did the scarecrow win an award? Because he was outstanding in his field! Now, let's calculate the rate at which the area changes with respect to the radius.
We are given that the area A of a circular water ripple is given by A = πr^2, where r is the radius. To find the rate at which the area changes with respect to the radius, we need to take the derivative of A with respect to r.
Using the power rule of differentiation, we get dA/dr = 2πr.
Now, let's find the rate of change of the area when the radius is 4 ft. Plugging in r = 4 into the derivative expression, we have dA/dr = 2π(4) = 8π ft^2/ft.
So, when the radius is 4 ft, the rate at which the area changes with respect to the radius is 8π ft^2/ft. But let's be honest, this ripple is making waves in both math and comedy!
To find the rate of change of the area with respect to the radius, we need to take the derivative of the area formula with respect to the radius, which is dA/dr.
Given that the area of the circular water ripple is A = πr^2, where r represents the radius, we can differentiate A with respect to r using the power rule for differentiation.
So, dA/dr = 2πr.
To find the rate of change of the area when r = 4ft, we substitute r = 4 into the expression for dA/dr:
dA/dr = 2π(4)
= 8π.
Therefore, when r = 4ft, the rate of change of the area with respect to the radius is 8π square feet per foot or 8π ft^2/ft.
To find the rate at which the area A changes with respect to the radius r, we need to differentiate the area formula with respect to r, which is:
A = πr^2
To differentiate, we apply the power rule. The power rule states that if we have a variable raised to a power, we can bring down the power in front and decrease the power by 1. Applying the power rule to differentiate A with respect to r:
dA/dr = 2πr
This equation gives us the rate at which the area changes with respect to the radius for any value of r.
To find the specific rate at r = 4ft, we substitute that value into the equation:
dA/dr = 2π(4)
dA/dr = 8π
So, the rate at which the area changes with respect to the radius when r = 4ft is 8π square feet per foot.