A trough is 2 feet wide and 8 feet long with a cross-section that is triangular (equilateral). Water is being pumped in at 2 gallons per minute. How fast is the depth rising when it is 3 inches deep? How fast is the depth rising when it is 6 inches deep? When the depth is 1 foot, the inflow rate changes. If it is rising at 1 foot per minute at that time, what is the new rate that water is being poured into the trough? [You’ll need to use the fact that 1 cubic foot is approximately 7.48 gallons.]
To solve this problem, we need to relate the dimensions of the trough to the volume of the water and find the rate at which the depth is changing.
First, let's find the formula for the volume of water in the trough. Since the cross-section is an equilateral triangle, the depth (d) is the height of the triangle. The base of the triangle is 2 feet, so the area of the base (A) is (sqrt(3)/4) * 2^2 = sqrt(3) square feet.
The volume (V) of the trough is then given by the formula V = A * d.
To find how fast the depth is changing with respect to time (ddt), we can use implicit differentiation, treating d as a function of time. So, ddt(V) = ddt(A * d).
Next, let's convert the volume from cubic feet into gallons. We'll use the conversion factor that 1 cubic foot is approximately 7.48 gallons. So, the volume V is in gallons, given by V = 7.48 * V(ft^3).
Therefore, ddt(V) = ddt(7.48 * V(ft^3)).
Now, we can solve the problem using the provided information:
1. When the depth is 3 inches (0.25 feet):
- Substitute d = 0.25 into the equation V = sqrt(3) * d to find the volume V.
- Differentiate both sides of the equation to find ddt(V).
- Plug in the values obtained to find the rate at which the depth is rising.
2. When the depth is 6 inches (0.5 feet):
- Repeat the same process as above to find the rate at which the depth is rising.
3. When the depth is 1 foot (12 inches):
- Find the new rate at which the depth is rising using the given information that it is rising at 1 foot per minute.
4. If the depth is 1 foot and the inflow rate changes:
- Convert the value of 1 cubic foot into gallons.
- Multiply the inflow rate by the conversion factor to find the new rate at which water is being poured into the trough.
Using these steps, we can calculate the answers to the given questions.