Lesson 16
Applying Volume and Surface Area
Let's explore things that are proportional to volume or surface area.
Problem 1
A landscape architect is designing a pool that has this top view:
![An irregular pentagon. Horizontal line, 9 feet. From the right end, slant down and right, 10 point 7 feet, down 1 point 5 feet, left 11 feet, up 12 feet to reach left end of original line.](https://cms-im.s3.amazonaws.com/fZzZQTW1RZssCXkJQAWL7Nsj?response-content-disposition=inline%3B%20filename%3D%227-7.7.newPP.trapezoidpooltop.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.newPP.trapezoidpooltop.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001503Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=748ac55a99523646514a7e5950e55506905915ac4cc471139c23ba943bee5f9e)
- How much water will be needed to fill this pool 4 feet deep?
- Before filling up the pool, it gets lined with a plastic liner. How much liner is needed for this pool?
- Here are the prices for different amounts of plastic liner. How much will all the plastic liner for the pool cost?
plastic liner (ft2) cost ($) 25 3.75 50 7.50 75 11.25
Problem 2
Shade in a base of the trapezoidal prism. (The base is not the same as the bottom.)
- Find the area of the base you shaded.
- Find the volume of this trapezoidal prism.
![A prism, height 12. Each base is a trapezoid, parallel sides 8 and 5, non-parallel sides 4 and 5. The side with length 4 is perpendicular to the parallel sides.](https://cms-im.s3.amazonaws.com/TjazBvyzv8hQkgDiFHg2t3W2?response-content-disposition=inline%3B%20filename%3D%227.7.newPP.trapprism.02.png%22%3B%20filename%2A%3DUTF-8%27%277.7.newPP.trapprism.02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001503Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f59325a11980fdf9859f4d0e87a261649446b3f4670d4801c934bf76c26c8f01)
Problem 3
For each diagram, decide if \(y\) is an increase or a decrease of \(x\). Then determine the percentage that \(x\) increased or decreased to result in \(y\).
![Two tape diagrams of equal size. Top diagram, 4 parts, 3 blue, total x, 1 part white. Bottom diagram, solid yellow, y.](https://cms-im.s3.amazonaws.com/R6nugbkKd4hpwEWNAhv8g8qF?response-content-disposition=inline%3B%20filename%3D%227-7.4.4.revised.image.02.png%22%3B%20filename%2A%3DUTF-8%27%277-7.4.4.revised.image.02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001503Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=8f9565e1b96e493ed23bb40e39b560c6336f5736f38fa9604a4908a9c08983ee)
![Two tape diagrams of equal length. Top diagram, 5 parts, 3 blue, total x, 2 white. Bottom diagram, solid yellow, y.](https://cms-im.s3.amazonaws.com/FYt4xqzAbETVBwmbW1vuAcEx?response-content-disposition=inline%3B%20filename%3D%227-7.4.4.revised.image.02b.png%22%3B%20filename%2A%3DUTF-8%27%277-7.4.4.revised.image.02b.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001503Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f238dca3e2f10f27b3c3d8edd4fb1530d7c3eaf7bf71a34d3cce6c09fa809e50)
![Two tape diagrams. Top diagram, 3 parts total x. Bottom diagram the same size as two parts above, solid, y.](https://cms-im.s3.amazonaws.com/6mVdcJXQxsUxecwLp8g8giRp?response-content-disposition=inline%3B%20filename%3D%227-7.4.4.revised.image.02c.png%22%3B%20filename%2A%3DUTF-8%27%277-7.4.4.revised.image.02c.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001503Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=681c7427c570ac0fc6d2cf6aa25903164fb6a55150eb53e46179448ad42a060a)
![Two tape diagrams. Top diagram, 3 parts, total x. Bottom diagram same size as one part above, solid, y.](https://cms-im.s3.amazonaws.com/5fMVM3Gkcyggn9JRSdsexMf9?response-content-disposition=inline%3B%20filename%3D%227-7.4.4.revised.image.02d.png%22%3B%20filename%2A%3DUTF-8%27%277-7.4.4.revised.image.02d.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001503Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c0863b432664ed8a82d7987679a5df3c0a7c15850d39cc2972165e9b26fafe7e)
Problem 4
Noah is visiting his aunt in Texas. He wants to buy a belt buckle whose price is $25. He knows that the sales tax in Texas is 6.25%.
- How much will the tax be on the belt buckle?
- How much will Noah spend for the belt buckle including the tax?
- Write an equation that represents the total cost, \(c\), of an item whose price is \(p\).