Question

SCALCET8 3.9.026.MI. A trough is 10 ft long and its ends have the shape of isosceles triangles that are 5 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 13 ft3/min, how fast is the water level rising when the water is 5 inches deep

Answer 1

The answer is "0.624"

Explanation:

`b = 5x\\\\h = x\\\\ l = 10 \\`

Using formula:

`V = ((1)/(2))(b)(h)(l) \\\\`

`= (1)/(2) * (5x)* (x) * (10)\\\\= 5x* x * 5\\\\= 25x^2`

`(dV)/(dt) = 50x \ (dx)/(dt) \\\\\text{(where x represents the height in feet and}\ (dv)/(dt) = 13\ (ft^3)/(min))\\\\(dx)/(dt) = ((1)/(50))((1)/(x)) \ (dV)/(dt)\\\\`

When the water is 5 inches deep then:

`x = ((5)/(12))\ ft\\\\13 =50((5)/(12)) \ (dx)/(dt)\\\\13 * 12 =50(5) \ (dx)/(dt)\\\\(13 * 12)/(50 * 5) = (dx)/(dt)\\\\(156)/(250) = (dx)/(dt)\\\\ (dx)/(dt)= 0.624`