To find how fast the water level is rising, we need to find the rate of change of the volume of the water with respect to time.
Let's first find the equation that relates the volume of the water to the depth of the water at any given point.
Since the bottom of the pool is rectangular, we can consider two separate regions: the shallow end region and the deep end region.
The volume, V, of the shallow end region can be calculated using the formula for the volume of a rectangular prism: V = length x width x height. In this case,
Vshallow = (20 feet x 8 feet x h), where h is the depth of the water at the shallow end.
The volume, V, of the deep end region can also be calculated using the formula for the volume of a rectangular prism: V = length x width x height. In this case,
Vdeep = [(40 feet - 20 feet)x (8 feet - 3 feet) x h], where h is the depth of the water at the deep end.
We subtract the shallow end dimensions from the overall pool dimensions to get the dimensions of the deep end region.
To find the total volume of the water in the pool, we need to sum up the volumes of the shallow and deep end regions:
Vtotal = Vshallow + Vdeep
Now, let's differentiate the total volume equation with respect to time (t):
dVtotal/dt = d(Vshallow)/dt + d(Vdeep)/dt
Since the water is being pumped into the pool at a constant rate of 40 cubic feet per minute, we have:
dVtotal/dt = 40
We are asked to find how fast the water level is rising (dh/dt) when the depth at the deep end (h) is 3 feet.
To find dh/dt, we need to differentiate both Vshallow and Vdeep with respect to t.
Differentiating Vshallow with respect to time, we get:
d(Vshallow)/dt = (20 feet x 8 feet x (dh/dt)) = 160(dh/dt)
Differentiating Vdeep with respect to time, we get:
d(Vdeep)/dt = [(40 feet - 20 feet) x (8 feet - 3 feet) x (dh/dt)] = 20 x 5(dh/dt) = 100(dh/dt)
Now, let's substitute these values into our earlier equation:
40 = 160(dh/dt) + 100(dh/dt)
Combining like terms, we have:
40 = 260(dh/dt)
Finally, solving for dh/dt, we get:
dh/dt = 40/260 = 2/13 feet per minute
Therefore, the water level is rising at a rate of 2/13 feet per minute when the depth at the deep end is 3 feet.