The water level is rising at a rate of 0.0015 ft/min when the depth at the deepest point is 5 ft.
How to determine how fast the water level is rising
To determine how fast the water level is rising when the depth at the deepest point is 5 ft
Denote the depth at the shallow end as y (in feet) and the depth at the deepest point as x (in feet). The width of the pool is 20 ft, and the length is 40 ft.
Since the cross-section is a triangle, we can use the similar triangles property:
(x - y) / (9 - 3) = 40 / 20
Simplifying the equation:
(x - y) / 6 = 2
x - y = 12
Next, find the relationship between the volume of the pool and the depth x. The volume of a triangular prism is given by the formula:
V = (1/2) * base * height * length
In this case, the base is the width of the pool (20 ft), the height is the average of the depths at the shallow end (y) and the deepest point (x), and the length is 40 ft. Therefore, we can express the volume V as:
V = (1/2) * 20 * [(y + x)/2] * 40
Simplifying the equation:
V = 400 * (y + x)
Now, differentiate both sides of the equation with respect to time:
dV/dt = 400 * (dy/dt + dx/dt)
Given that the pool is being filled at a rate of 0.6
/min, we can substitute dV/dt with 0.6:
0.6 = 400 * (dy/dt + dx/dt)
Since we are interested in finding dx/dt when x = 5 ft, find dy/dt at that point as well.
From the similar triangles equation, we know that x - y = 12.
When x = 5, we can solve for y:
5 - y = 12
y = -7
Now, substitute the values into the equation:
0.6 = 400 * (dy/dt + dx/dt)
We also know that when x = 5, y = -7, so dy/dt = 0.
0.6 = 400 * (0 + dx/dt)
Solving for dx/dt:
dx/dt = 0.6 / 400
dx/dt = 0.0015 ft/min
Therefore, the water level is rising at a rate of 0.0015 ft/min (rounded to five decimal places) when the depth at the deepest point is 5 ft.
A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at is deepest point. A cross-section is shown in the figure. If the pool is being filled at a rate of 0.6 ft3/min, how fast is the water level rising when the depth at the deepest point is 5 ft? (Round your answer to five decimal places.)