Question

A trough is 15 ft long and 4 ft across the top, as shown in the figure. Its ends are isosceles triangles with height 3 ft. Water runs into the trough at the rate of 2.5 ft^3/min. How fast is the water level rising when it is 2 ft deep?

Answer 1

the width of the water surface when the depth is y is 4y/3

So, the cross-section at depth y has area
1/2 * y * 4y/3 = 2y^2/3

So the volume of water when the depth is y is

v = 10y^2
dv/dt = 20y dy/dt
at y=2,
5/2 = 40 dy/dt
dy/dt = 1/16 ft/min