Water is leaking out of a pool at a rate of 4 cubic inches per second, what is the rate of change of the height of the water at the instant the volume equals 20ft cubic feet
V= (l)(w)(h)
since the length and width do not change, only V and h.
dV/dt = (lw) dh/dt
We must have the same units,
20 cubic feet = 20(12^3) or 34560 cubic inches
when V = 34560 and dV/dt = -4
34560 = (lw) h
lw = 34560/h
-4 = 34560/h dh/dt
dh/dt = -h/8640 inches/second
Are you sure there were no other units given?
To find the rate of change of the height of water, we first need to convert the volume from cubic feet to cubic inches.
Since 1 ft = 12 inches, 1 cubic ft = 12^3 = 1728 cubic inches.
Therefore, 20 cubic ft = 20 * 1728 = 34,560 cubic inches.
Now, we know that the rate of change of the volume of water is 4 cubic inches per second, which means the rate of change of the volume is constant over time.
Let's assume the height of the water is h inches. The volume of the water in the pool can be calculated using the formula: V = A * h, where A is the area of the base of the pool.
As we are only concerned with the rate of change of the height, we can differentiate both sides of the equation with respect to time (t):
dV/dt = A * (dh/dt)
Since the area of the base of the pool (A) is constant, let's represent it as a constant, K. Therefore, we have:
dV/dt = K * (dh/dt)
Given that dV/dt is 4 cubic inches per second, and V is 34,560 cubic inches, we can substitute these values into the equation:
4 = K * (dh/dt)
To find the value of dh/dt (the rate of change of the height of the water), we need to rearrange the equation:
dh/dt = 4 / K
Now, we can find the value of K by using the formula for the area of a circle, where the radius is half the diameter of the pool. Let's assume the diameter of the circular pool is D. Therefore, the radius is D/2.
The area of the base of the pool (A) is given by: A = π * (D/2)^2
Since we only need the rate of change of the height of the water and not the specific measurements of the pool, we can represent the constant K as a general constant, C. Therefore, we have:
A = C
Now, we can substitute the value of K with C in the equation we derived earlier:
dh/dt = 4 / C
Therefore, the rate of change of the height of the water is 4 / C.