The radius is increasing fastly at a rate of 2.75 feet per second when it is already 30 feet.
Here, it has been told that the rate of change of the area with respect to the time is = 520 ft²/sec - (i)
This can also be written as = dA/dt (ii)
We need to find the rate of change of radius with respect to the time at 30 feet. This can be written as = dr/dt (iii)
We already know that the area of a circle can be found out using the formula = πr² (iv)
Differentiating equation (ii) with respect to time, we get -
= d(πr² )/dt
= π*(dr²/dt)
= π*(dr²/dr)*(dr/dt)
= π*(2r)*(dr/dt)
= dr/dt = (dA/dt)*(1/ π*2r)
Using the given values, we get -
= (dr/dt) = 520 / π*2*30ft
= (dr/dt) = 2.75 ft/sec
Henceforth, we find that the rate of change of the radius with respect to time at 30 feet is going to be 2.75 ft/sec.
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