Yang G. answered • 11/18/19

Data Scientist specializing in STEM subjects

First let's find the Volume of the trough, I assume it's an inverted triangle where the sides are equal since the diagram is not shown. All units match so I will work with numbers only.

**Volume of a triangle prism is given by:**

V = (b * h )/2 * l

Length or l is a constant at 9, so we can sub it in the equation.

V = 9/2 * (b * h)

**Substitute out base**

Base and height of the filled trough at any time is related by trigonometry. We're interested in height so we will substitute out base with height.

This part would be a lot easier with a diagram, but unfortunately I can't upload one. Using properties of isosceles right triangle and tan:

tan(45 degrees) = h/(b/2)

b = 2h

V = 9/2 *(2h * h) = 9h^2

**Rate**

Take derivative of volume with respect to time to find equation for rate of filling the trough.

dV/dt = 2 * 9 *h dh/dt = 18h dh/dt

re-arranging for dh/dt

dh/dt = dV/dt /(18h)

Subbing in dV/dt = 2

dh/dt = 2/(18 h) = 1/(9h)

**Find h at 2 min**

V = 9h^2

V = t * rate = 2 * 2 = 4

4 = 9h^2

re-arranging for h:

h = sqrt(4/9) = 2/3

**Subbing in h to rate equation**

dh/dt = 1/(9h) = 1/(9 * 2/3) = 1/6

The rate of change for height is 1/6 cubic feet/min at 2 min.