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??
The area of the "disk" cannot be defined, because the height is missing. The area will be assumed to mean area of the circle.
A=πr²
dA/dt=dA/dr*dr/dt=2πr*2 cm²/s
A)A=Пr^2
B)dA/dt=dA/dr*dr/dt=2Пr*dr/dt
C)dA/dt=2П*7*2cm^2/s=87.96cm^2/s
All correct!
thank you!
A) The relation between the area of a disk (A) and its radius (r) is given by the formula A = πr^2. This formula states that the area of a disk is equal to the constant value π multiplied by the square of its radius.
B) To find the derivative of the relation between A and r with respect to time (t), we need to apply the chain rule. The derivative of A with respect to t (dA/dt) is equal to the derivative of A with respect to r (dA/dr) multiplied by the derivative of r with respect to t (dr/dt).
Since A = πr^2, we can find the derivative of A with respect to r by differentiating the equation A = πr^2 with respect to r. This gives us dA/dr = 2πr.
Thus, the derivative of the relation between A and r with respect to time (t) is dA/dt = (dA/dr) * (dr/dt) = 2πr * (dr/dt).
C) When r = 7cm and dr/dt = 2 cm/s, we substitute these values into the previously derived expression for the derivative:
dA/dt = 2πr * (dr/dt)
= 2π(7) * (2)
= 28π cm^2/s
Therefore, when the radius is 7cm and the rate at which the radius is growing is 2 cm/s, the rate at which the area is growing is 28π cm^2/s.