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%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T011149Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=d6184ff5e00c73189be953603db6b481b082846193a638a352a4aa677f8bff7a)
- 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%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T011149Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=72726c6e6c11749d6a824ea0b712790343080fde75a21482efe58d3ca330fda5)
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%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T011149Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=16b104c5d77ab9f066a7bb6732d17b6c60ca457fdb9ebd556f453f6ca3b944e4)
![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%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T011149Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=fa2cae51400441d1b74fb536c5043c44075a9ce4f64ea8c8d9fdcfec158bf60d)
![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%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T011149Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=93f16d2c411da67da6843cdb880c851611857f23da7ebfba0029689f287ef453)
![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%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T011149Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c64ad3f22ea7491ca279cc79ed9eb23b059a11189dcdab52be9328b6c8aca801)
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 \(p\).