The radius of a disk grows at 2cm/s. Calculate the growth rate of the disks area when the radius mesures 7 cm. (Designate the area of the disk with A and the radius with r)
A) Find the relation between A and r
B) Find the derivative of this relation on time t
dA/dt =
C) When r = 7cm and dr/dt = 2 cm/s, find dA/dt
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I'm just not sure for A, is the relation going to be dA/dt = dr/dA * dr/dt??
To solve this problem, let's go step by step:
A) Find the relation between A and r:
The area of a disk is given by the formula A = πr^2. Therefore, the relation between the area (A) and the radius (r) is A = πr^2.
B) Find the derivative of this relation on time t (dA/dt):
To find the derivative of A with respect to t (dA/dt), we need to differentiate both sides of the equation A = πr^2 with respect to t.
Since A is a function of r, we can use the chain rule to differentiate A with respect to t:
dA/dt = dA/dr * dr/dt
Derivative of A with respect to r (dA/dr):
Differentiating A = πr^2 with respect to r will give us 2πr.
Derivative of r with respect to t (dr/dt):
Given that the radius is growing at a constant rate of 2 cm/s, we have dr/dt = 2.
Therefore, dA/dt = 2πr * 2 = 4πr.
C) When r = 7 cm and dr/dt = 2 cm/s, find dA/dt:
Substituting the values r = 7 cm and dr/dt = 2 cm/s into the expression for dA/dt, we get:
dA/dt = 4π(7) = 28π cm^2/s
So, when the radius measures 7 cm and is growing at 2 cm/s, the growth rate of the disk's area is 28π cm^2/s.