Lesson 13
Decomposing Bases for Area
Let’s look at how some people use volume.
Problem 1
You find a crystal in the shape of a prism. Find the volume of the crystal.
The point \(B\) is directly underneath point \(E\), and the following lengths are known:
- From \(A\) to \(B\): 2 mm
- From \(B\) to \(C\): 3 mm
- From \(A\) to \(F\): 6 mm
- From \(B\) to \(E\): 10 mm
- From \(C\) to \(D\): 7 mm
- From \(A\) to \(G\): 4 mm
![An irregular pentagonal prism with base A, F, E, D, C. Segment A, G indicates the height of the prism. Point B lies between A and C.](https://staging-cms-im.s3.amazonaws.com/wauRgHo3jjdirSvGxuDHPCkR?response-content-disposition=inline%3B%20filename%3D%227-7.6.C.PP.Image.06.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.C.PP.Image.06.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T082427Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=8fefd8aa03851d69aa20245b39c8c3b7e42c8f1aae99d495a42f398230e315c5)
Problem 2
A rectangular prism with dimensions 5 inches by 13 inches by 10 inches was cut to leave a piece as shown in the image. What is the volume of this piece? What is the volume of the other piece not pictured?
![A right trapezoidal prism. Each base is a trapezoid with bases 13 inches and 1 inch, height 5 inches. The prism has height 10 inches.](https://staging-cms-im.s3.amazonaws.com/pcSxM4jR9ZY3h51jxUssoQcs?response-content-disposition=inline%3B%20filename%3D%227-7.6.C.PP.Image.05.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.C.PP.Image.05.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T082427Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=50e8939355beace891af5e4e1d0e863c735a64c883a86703d575c0fc8f510b5b)
Problem 3
A triangle has one side that is 7 cm long and another side that is 3 cm long.
-
Sketch this triangle and label your sketch with the given measures. (If you are stuck, try using a compass or cutting some straws to these two lengths.)
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Draw one more triangle with these measures that is not identical to your first triangle.
- Explain how you can tell they are not identical.
Problem 4
Select all equations that represent a relationship between angles in the figure.
![Three points intersect to form 6 lines. Clockwise, the angles measure b degrees, 30 degrees, 90 degrees, a, degrees, c degrees, blank.](https://staging-cms-im.s3.amazonaws.com/inij2gqPDAFmmhmEdBrfdbr8?response-content-disposition=inline%3B%20filename%3D%227-7.7.A4.new.PP.04.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.A4.new.PP.04.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T082427Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=14d99b92d4b60838c63f681b26475737c825e21c20c02b9696b93d0858de7fa0)
\(90-30=b\)
\(30+b=a+c\)
\(a+c+30+b=180\)
\(a=30\)
\(a=c=30\)
\(90+a+c=180\)
Problem 5
A mixture of punch contains 1 quart of lemonade, 2 cups of grape juice, 4 tablespoons of honey, and \(\frac12\) gallon of sparkling water. Find the percentage of the punch mixture that comes from each ingredient. Round your answers to the nearest tenth of a percent. (Hint: 1 cup = 16 tablespoons)