A cicular metal plate is being heated and expands so that its radius increases at the rate of 0.25 mm/min. How fast is the area increasing when the radius is 10 cm?
To find the rate at which the area is increasing, we need to differentiate the equation for the area of a circle with respect to time and then substitute the given values.
The equation for the area of a circle is A = πr^2, where A is the area and r is the radius.
Differentiating both sides of this equation with respect to time (t) gives:
dA/dt = 2πr(dr/dt)
In this case, we are given that dr/dt = 0.25 mm/min, and we need to find dA/dt when r = 10 cm.
First, let's convert the given values into the appropriate units:
dr/dt = 0.25 mm/min = 0.25/10 = 0.025 cm/min
r = 10 cm
Now, we can substitute these values into the equation:
dA/dt = 2π(10)(0.025) = 0.5π cm^2/min
Therefore, the area is increasing at a rate of 0.5π cm^2/min when the radius is 10 cm.