# Derivatives, rates of change (trapezoidal prism)

1. Dec 2, 2013

### physics604

1. A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2 m^3/min, how fast is the water level rising when the water is 30 cm deep?

2. Relevant equations
$$V=\frac{(a+b)h}{2}×L$$
3. The attempt at a solution

Given (I converted all to cm):
$$a=30$$ $$\frac{da}{dt}=0$$ $$base=80$$ $$height=50$$ $$\frac{dV}{dt}=0.2$$ $$h=30$$ $$L=10$$ $$\frac{dL}{dt}=0$$

I need to find $$\frac{dh}{dt}$$

After I find the derivative of V using natural log.
$$lnV=\frac{1}{2}ln(a+b)hL$$
$$lnV=\frac{1}{2}ln(a+b)+lnh+lnL$$
$$\frac{dV}{dt}=\frac{1}{2}(\frac{db/dt}{a+b}+\frac{dh/dt}{h})×V$$

I don't know how I'm supposed to find $$\frac{db}{dt}$$ i get b=60.

Any help is much appreciated.

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Last edited: Dec 2, 2013
2. Dec 2, 2013

### Simon Bridge

This is the same as the other one - you don't need to deal with all those logs.

(a+b)L/2 is just a constant that you are given. Put k=2/(a+b)L
Lynchpin You need to express b in terms of h.

Think it through - let a is the bottom, and c is the top, the overall height is H.
0<h<H
when h=0, what is b(h) equal to?
when h=H, what is b(h) equal to?
what formula makes that happen? b(h)=

* Rearrange your formula to make h the subject.
* differentiate both sides to get dh/dt in terms of dV/dt.
* You are given dV/dt.

Last edited: Dec 2, 2013
3. Dec 2, 2013

### physics604

Thanks, but I still get this.

$$2V=(a+b)hL$$ $$h=\frac{2V}{(a+b)L}=Vk$$ $$\frac{dh}{dt}=\frac{dV}{dt}k=0.2×\frac{2}{(30+60)1000}$$

I don't get the right answer, which is 10/3.

4. Dec 2, 2013

### Simon Bridge

Still the wrong formula.
I misread you - I thought you put b as the top width - my bad.
I have rewritten post #2 to reflet this.

Think it through - let a is the bottom, and c is the top, the overall height is H.
0<h<H
when h=0, what is b(h) equal to?
when h=H, what is b(h) equal to?
what formula makes that happen? b(h)=

5. Dec 2, 2013

### Simon Bridge

Still the wrong formula.
I misread you - I thought you put b as the top width - my bad.
I have rewritten post #2 to reflet this.

Think it through - let a is the bottom, and c is the top, the overall height is H.
0<h<H
when h=0, what is b(h) equal to?
when h=H, what is b(h) equal to?
what formula makes that happen? b(h)=

6. Dec 2, 2013

### SteamKing

Staff Emeritus
Watch your units!

If you are going to use cm for the dimensions of the trough, then the fill rate of 0.2 m^3/min should also be converted to units of cm^/min.

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