M2L3 PRACTICE Let’s build a pool! By M Willatt The pool picture used in this presentation is “Pool for Skipball” by Water Cooler Sports on commons.Wikimedia.org via CC BY 1.0 Graphs made on TI-83 calculator.
Let’s design a rectangular pool for my backyard! Since I’m a math teacher, I’m fine with odd dimensions as long as we can maximize the volume. I got a great deal on pool epoxy paint, but I only have enough cover 440 square foot. This will be painted inside the pool to seal it.
“Pool for Skipball” by Water Cooler Sports on commons.Wikimedia.org via CC BY 1.0
Since we don’t know the dimensions yet, let’s use x to represent the width of my new pool in feet.
Note: Please know that due to the perspective we are looking at the pool even though the two x’s appear to be different lengths, they are actually the same length.
I would like the length to be double the width so we can use 2x to represent that.
We’ll use d to represent the depth in feet.
We know how much surface area the paint can cover 440 ft2. Keep in mind, this is like an open top box problem as we would only be painting the sides and bottom of the pool. You can’t paint the top
First, let’s break down the area for the left and right sides of the pool: Notice each would have the same area of 2xd.
Next we’ll focus on the area for the closest and farthest sides of the pool:
Again the front and back would both have an area of xd.
The only part that would still need paint is the bottom of the pool.
Lastly, we need to add up all those surfaces to get the total surface area: +
+
Total Surface Area = 2 sides (left & right) + 2 sides (front & back) + bottom = 2(2xd) + 2(xd) + 2x 2 = 4xd + 2xd + 2x 2 = 6xd + 2x2
We know the Total S.A. possible is 440‌ 440 = 6xd + 2x2
Now to find volume, you probably already remember we need to multiply our 3 dimensions: V = x * 2x * d We need to eliminate a variable so we can graph to find the max volume. Let’s replace the d with an equivalent expression.
X min and Ymin should not be below 0 as my volume (y) and width (x) cannot be negative.
volume
I tried a few numbers for x max and y max until I got a window size I liked.
pool width
Try finding the maximum on your calculator.
volume
maximum
pool width
1. 2. 3. 4.
If you forgot how, press 2ND, TRACE,4. Then move your cursor to the left side of the max and hit ENTER. Next, move the cursor to the right side of the max and hit ENTER. When it says “Guess?” hit ENTER again.
Did you get (8.56, 837.32) ?
volume
maximum
pool width