calculus help... related rates...what am i doing wrong?

Question

a trough is 5 ft long and its vertical cross sections are inverted isosceles triangles w/ base 2 ft and height 3 ft. wather i being siphoned out of the trough at the rate of 2 cubic feet/min. At any time t, let h be the depth and V be the volume of the water in the trough

-what is the rate of change in h at the instant when the trough is 1/4 full by volume?

-what is the rate of change in the area of the surface of the water at the instant when the trough is 1/4 full by volume

so i know i have to find dV/dt in terms of h, and then plug in a value for h once i've done that. the problem is when i find the derivative i don't get a variable h, i get 5(dh/dt). also, i don't know what to plug in for h, whether it's 15/4 since the volume is 15 cubic feet and 1/4 of it would be 15/4. or do i plug in 3/4 since the height is 3 ft.

any help would be appreciated
thanks!

Answer 1

volume=length into area of cross section
=5*1/2*2*3=15cft

and at any time b/h=2/3
or h=3b/2 or b=2h/3
volume=l*1/2*bh=(1/2)*5*(2h/3)*h
when the volume id 1/4th
15/4=(1/3)h^2
find h=3rt5/4

Answer 2

I'll give you hand with this.

Ok, you have the cross-section area of the triangle... A=(1/2)bh
so the volume is: V=(1/2)bh(5ft)

As the volume decreases, both h and b change... you just need to find the relationship of the two. the ratio will always stay the same, so we know that b=(2/3)h
So then,
V=(5/3)(h^2)

dV/dt=(10/3)(h)(dh/dt)

So, now we have the two equations we need for part one.

first, find what the volume would be if the trough were full.
V=(5/3)(9ft^2)=15ft^3

when it is 1/4 full, the volume is (15ft^3)/4.
therefore, the height is 1.5 ft.

so, we get the following equation:

2ft^3/s=(10/3)(1.5ft)(dh/dt)

use this formula to solve for dh/dt. that's your answer for the first part. you now need to use it for part 2.

The surface area is:

A=b*5ft
A=(2/3)*h*5ft
dA/dt=(2/3)(dh/dt)(5ft)

So, sub in your answer from the first part, and you get the answer for part two.


I hope this helps.