This related rates problem is killing me, help please?

[pitt14]

Homework Statement

A pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. If the pool is shaped as shown (I attached a picture), and is being filled at a rate of 8 ft^3/min, how fast is the water level rising when the depth at the deepest point is 5ft?

Homework Equations

I *think* I need the volume of a trapezoidal prism?
V = Bh = (.5*a*b*h)*w
I am completely lost with the rest of this

The Attempt at a Solution

All I can figure out is that V=2700. I need a lot of help with this.

 

[Mark44]

Homework Statement

A pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. If the pool is shaped as shown (I attached a picture), and is being filled at a rate of 8 ft^3/min, how fast is the water level rising when the depth at the deepest point is 5ft?

Homework Equations

I *think* I need the volume of a trapezoidal prism?
V = Bh = (.5*a*b*h)*w
I am completely lost with the rest of this

The Attempt at a Solution

All I can figure out is that V=2700. I need a lot of help with this.

Is 2700 ft³ the volume when the pool is filled? If so, that won't be any use in this problem.

What you need to do is relate the volume V and the height of water h, then differentiate to get dV/dh.

The volume at a given level will be the cross-sectional area times 20, the width of the pool. The lower base of the trapezoid is 12 ft. The upper base will be 12 plus the contributions of the two triangular pieces. The one on the left has an altitude of 6' and a base of 6', so when the water is h ft deep, the contribution to the upper base of the trapezoid will also be h ft, using similar triangles. I'll leave figuring out the contribution of the triangle on the right to you.

 

[pitt14]

Is 2700 ft³ the volume when the pool is filled? If so, that won't be any use in this problem.

What you need to do is relate the volume V and the height of water h, then differentiate to get dV/dh.

The volume at a given level will be the cross-sectional area times 20, the width of the pool. The lower base of the trapezoid is 12 ft. The upper base will be 12 plus the contributions of the two triangular pieces. The one on the left has an altitude of 6' and a base of 6', so when the water is h ft deep, the contribution to the upper base of the trapezoid will also be h ft, using similar triangles. I'll leave figuring out the contribution of the triangle on the right to you.

Thank you. I am still really confused, though. I can't figure out how to set this up. I figured out the perimeter of both triangles in the trapezoid. But what do I use for the volume equation if I don't use the volume of the full pool?

 

[Mark44]

The volume of the pool when it's full is of no use whatsoever in this problem. As I already said, you need to get an equation for the volume V as a function of h, the height of water at a given time.

The perimeters of the triangles aren't relevant either. The length of the upper base of the trapezoid is 12 + A + B, where A is the length of the base of the triangle on the left and B is the length of the base of the triangle on the right.

When you get V as a function of h, differentiate to get V'(h), and then evaluate at h = 5.

 

[pitt14]

The volume of the pool when it's full is of no use whatsoever in this problem. As I already said, you need to get an equation for the volume V as a function of h, the height of water at a given time.

The perimeters of the triangles aren't relevant either. The length of the upper base of the trapezoid is 12 + A + B, where A is the length of the base of the triangle on the left and B is the length of the base of the triangle on the right.

When you get V as a function of h, differentiate to get V'(h), and then evaluate at h = 5.

This is the part I can't figure out. I don't know how to get V as a function of h. I have been trying this problem for a week now. Can I get some guidance as to how to get this?

 

[Mark44]

Read what I wrote in my two previous posts.