related rates problem

Question

A trough is 14 ft long and its ends have the shape of isosceles triangles that are 4 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 7 inches deep?

v=(1/2)bhl

v=(1/2)(4h)(h)(14)

v=28h^2

dv/dt=56h(dh/dt)

h=7/12

(13)=56(7/12)(dh/dt)

dh/dt=0.397

why is this wrong?

Answer 1

I think your answer needs to remember that when the water rises BOTH the h and the b are changing (the l is constant). Therefore, your derivative needs to use the product rule, i.e.

dVdt=(L/2)[b dh/dt + h dh/dt]

Then you need to remember that b and h are related by b = 4h so that db/dt=4 dh/dt.

This should get you the answer you need. If not, get back to me!