Let A be the area of the circle with the radius r. if dr/dt = 5, find dA/dt when r = 1
To find dA/dt, the rate at which the area of the circle is changing with respect to time, we need to use the chain rule of differentiation.
First, we have the formula for the area of a circle: A = πr^2.
Taking the derivative of both sides with respect to time (t), we get:
dA/dt = d/dt (πr^2)
To apply the chain rule, we need to consider that r is a function of t (r = g(t)). Since we are given that dr/dt = 5, we can rewrite the formula as:
dA/dt = (dA/dr) * (dr/dt)
Now, we can differentiate the formula for A = πr^2 with respect to r:
dA/dr = d/dt (πr^2)
The derivative of A with respect to r is: 2πr.
Substituting the known values into the equation, we have:
dA/dt = (2πr) * (dr/dt)
When r = 1 and dr/dt = 5:
dA/dt = (2π * 1) * 5
dA/dt = 10π
Therefore, when r = 1 and dr/dt = 5, dA/dt is equal to 10π.
A = pi r^2
dA/dt = pi * 2r dr/dt
You know r and dr/dt, so ...