when a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm/min. At what rate is the plate's area increasing when the radius is 60 cm?
area=PI*r^2
d area/dt= PI 2r dr/dt
solve for d Area/dt, given r=60cm, dr/dt=.01cm/min
To find the rate at which the plate's area is increasing, we need to use the formula for the area of a circle:
A = πr^2
where A is the area and r is the radius.
We are given that the radius is increasing at a rate of 0.01 cm/min. Let's denote this rate as dr/dt.
We need to find the rate at which the area is changing, or dA/dt.
To do this, we can differentiate the area formula with respect to time (t):
dA/dt = d/dt (πr^2)
Using the chain rule, we have:
dA/dt = 2πr(dr/dt)
Substituting the given values, we have:
dA/dt = 2π(60 cm)(0.01 cm/min)
Calculating this expression, we get:
dA/dt = 2π(0.6 cm/min)
Finally, we can find the numerical value of dA/dt:
dA/dt ≈ 3.77 cm²/min
Therefore, when the radius is 60 cm, the plate's area is increasing at a rate of approximately 3.77 cm²/min.