Swimming Pool Volume and Related Rates

In summary, the water level is rising at a rate of .8 ft^3/minute when the depth at the deepest point is 5 feet.
  • #1
orbital92
2
0

Homework Statement


A swimming pool is 20 feet wide, 40 feet long, 3 feet deep at the shallow end and 9 feet deep at its deepest point. If the pool is being filled at a rate of 0.8 feet^3/min, how fast is the water level rising when the depth at the deepest point is 5 feet?

200710191446396332840199968500003988.jpg

Homework Equations



I believe the formula for volume of a swimming pool is V = lw((h+H/2)) where h is the shallowest depth and H is the deepest depth? Not 100% sure

The Attempt at a Solution



I know that dV/dt = 0.8 ft^3/minute. The dimensions of the pool (20 by 40 ft) are constant and do not change. h can be expressed in terms of H (H/3) and subbed into the equation. I need to find dH/dt when H=5 feet. Other than this I'm completely stuck! Thanks for any help!
 
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  • #2
The dimensions of the pool are actually NOT constant. Yes, the width stats at 20 feet, but the length actually depends on how much water is in there. For instance, before the water reached 6ft of depth, its length approaches 34 feet (6+12+16). At 6ft of depth, you switch over to the "shallow end", and it immediately jumps to the full 40ft length.
I'll take a look at this and get back to you.
(edit...)
Ok, we're in business.
The "big picture" is that we want to express the volume in terms of just ONE other variable. Otherwise, we'd have to use the product rule and we'd need more information, etc...

There may be an easier way to do this, but ... well, nobody has posted it yet!

Let's scrap the idea of V = L*W*H.
The volume of a prism is the area of the base * (another dimension = "width" in our case).
The "base" is the trapezoid that we see in the picture. The area of this depends on how deep the water is (obviously).

I'm actually going to break-up the trapezoid into three parts:
a) left triangle. This is the blue region directly above the "6ft" segment.
b) rectangle of width 12ft, above the "12ft" segment
c) triangle on the right, above the "16ft" segment.

The area of the base is the sum of these three parts. Let's denote the depth of the water by "h".

a) The dimensions of this triangle are x1 and h. This triangle is similar to a 6x6 triangle (since those are the dimensions of the maximum length and height of this triangle). So we setup the proportion x1/h = 6/6. So x1=h.
The area of this triangle is .5(length)*(height) = .5(x1)*(h) = .5(h)(h) = .5h2.
b) The area is 12*h
c) Use similar triangles to determine the area as a function of h ONLY.

Add these up. You will get an expression in terms of h ONLY.
Take the derivative. Don't forget the chain rule (which will give you a factor of dh/dt).
Plug.
Chug.
 
Last edited:
  • #3
This helped me solve it correctly! Thank you The Chaz :D
 

Related to Swimming Pool Volume and Related Rates

What is a related rates word problem?

A related rates word problem is a type of mathematical problem that involves finding the rate of change of one quantity with respect to another, while both quantities are changing at the same time. These problems typically involve multiple variables and require the use of calculus to solve.

How do you approach solving a related rates word problem?

To solve a related rates word problem, it is important to first identify all of the given information and the variables that are changing. Then, use the given information to create an equation that relates the changing variables. From there, use calculus techniques to differentiate the equation with respect to time and solve for the desired rate of change.

What are some common real-life situations that can be modeled as related rates word problems?

Related rates word problems can be used to model various situations in real life, such as the rate at which the water level in a pool is changing, the speed of a car approaching a traffic light, or the growth rate of a population. These types of problems are useful in fields such as physics, engineering, and economics.

What are some tips for solving related rates word problems?

Some tips for solving related rates word problems include drawing a diagram to visualize the problem, clearly labeling all given information and variables, and being consistent with units of measurement. It is also helpful to check your answer and make sure it makes sense in the context of the problem.

What are some common mistakes to avoid when solving related rates word problems?

One common mistake to avoid when solving related rates word problems is not carefully reading and understanding the given information. It is important to correctly identify which variables are changing and which are constant. Another mistake to avoid is not being consistent with units of measurement, as this can lead to incorrect answers. It is also important to double check your calculations and make sure they are accurate.

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