a stone is thrown into a pond creating ripples that are concentric circles. the rate of change of the radius of the circle is 3 ft/min. if the radius is 0 at the time=0, how fast is the area increasing after 4 minutes
Radius = r
Area = 4π(r^2)
As per the question: dr/dt = 3ft/min
dr/dt = 3
When t = 4, r = t*3 = 4*3 = 12 ft
d(Area)/dt = d(4π*r^2)/dt
= 4π*2r*(dr/dt)
Plug in r = 12, dr/dt = 3, you'll get the answer (in feet squared per minute)
To find out how fast the area is increasing, we need to differentiate the formula for the area of a circle with respect to time. The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
Differentiating both sides of the equation A = πr^2 with respect to time (t), we get dA/dt = 2πr(dr/dt).
Given that dr/dt = 3 ft/min and we want to find how fast the area is increasing after 4 minutes, we can substitute these values into the equation above.
dA/dt = 2πr(dr/dt)
dA/dt = 2π(0 + 3(4))
dA/dt = 2π(0 + 12)
dA/dt = 24π
Therefore, after 4 minutes, the area is increasing at a rate of 24π square feet per minute.