Answer: 5 / (36π) ft/sec.
Step-by-step explanation: To find how fast the height of the water in the pool is increasing, we need to use the formula for the volume of a cylinder.
The formula for the volume of a cylinder is V = πr^2h, where V is the volume, r is the radius, and h is the height.
Given that the radius of the pool is 6ft and the pool is being filled at a rate of 5ft^3/sec, we can find how fast the height of the water is increasing by differentiating the volume equation with respect to time.
Differentiating both sides of the equation V = πr^2h with respect to time (t), we get:
dV/dt = d(πr^2h)/dt
The derivative of the volume with respect to time (dV/dt) represents the rate at which the volume is changing over time, which is given as 5ft^3/sec.
Now, let's substitute the given values into the equation:
5 = π(6^2)(dh/dt)
Simplifying this equation, we get:
5 = 36π(dh/dt)
To find how fast the height of the water (dh/dt) is increasing, we can solve for it:
dh/dt = 5 / (36π)
Therefore, the height of the water is increasing at a rate of 5 / (36π) ft/sec.