When a circular plate of metal is heated in an oven, its radius increases at a rate of 0.02 cm/min. At what rate is the plate's area increasing when the radius is 100 cm?
Answer = _______ cm^2/min
cug
To find the rate at which the plate's area is increasing, we'll need to use the formula for the area of a circle and the given information about the rate of change of the radius.
The formula for the area of a circle is A = πr^2, where A represents the area and r represents the radius.
Since we're given the rate of change of the radius, we can differentiate the area formula with respect to time (t) to find the rate at which the area is changing.
dA/dt = 2πr(dr/dt)
Here, dA/dt represents the rate of change of the area, dr/dt represents the rate of change of the radius, and π is a constant.
Plugging in the values we know:
rate of change of the radius (dr/dt) = 0.02 cm/min
radius (r) = 100 cm
dA/dt = 2π(100)(0.02)
To calculate the final answer, we can plug in the values and solve:
dA/dt = 2π(100)(0.02)
= 4π cm^2/min
Therefore, when the radius is 100 cm, the plate's area is increasing at a rate of 4π cm^2/min.