Lesson 11
Slicing Solids
Let's see what shapes you get when you slice a three-dimensional object.
Problem 1
A cube is cut into two pieces by a single slice that passes through points \(A\), \(B\), and \(C\). What shape is the cross section?
![A cube is indicated. Point A is located on the back, top right vertex, Point B is located on the front, top left vertex, and Point C is located on the front, bottom left vertex](https://cms-im.s3.amazonaws.com/7a3HJuWgnzTbhQHwgUATtXdK?response-content-disposition=inline%3B%20filename%3D%227-7.6.C.PP.Image.02.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.C.PP.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=20240718T010952Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=62313ebbfd31256986be90ae80b68d746d99543b9f0c15ace8872a8cf98c8617)
Problem 2
Describe how to slice the three-dimensional figure to result in each cross section.
Three-dimensional figure:
Cross sections:
![A pyramid, the base of which is a triangle.](https://cms-im.s3.amazonaws.com/yudAeQVTcUJ9fvJaNYpwVvLE?response-content-disposition=inline%3B%20filename%3D%227-7.7.newPP_Image_6ghi.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.newPP_Image_6ghi.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=20240718T010952Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=48e15b3abfd9b9f664b8dd297421948dbcbd575b102a43b2f61f18248a17aa58)
![Two figures, a triangle and a trapezoid.](https://cms-im.s3.amazonaws.com/BNYkWEpmyXcxhg15yG2v2NxQ?response-content-disposition=inline%3B%20filename%3D%227-7.7.newPP_Image_7jkl.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.newPP_Image_7jkl.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=20240718T010952Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=1345a4892e11c17098f8e70d971417c9c67e79855e27ee37a1493c520f1aa2e1)
Problem 3
Here are two three-dimensional figures.
![Two three-dimensional figures. Figure A a triangular prism. Figure B is a triangular pyramid.](https://cms-im.s3.amazonaws.com/y1QdBS1CE3t74VFx5t8p12Mz?response-content-disposition=inline%3B%20filename%3D%227-7.7.newPP_Image_4abc.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.newPP_Image_4abc.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=20240718T010952Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ca81b4ee06192d04d2bfd59b3de468ec592560b9a46d802d61a3781591f6c294)
Describe a way to slice one of the figures so that the cross section is a rectangle.
Problem 4
Each row contains the degree measures of two supplementary angles. Complete the table.
measure of an angle | measure of its supplement |
---|---|
\(80^\circ\) | |
\(25^\circ\) | |
\(119^\circ\) | |
\(x\) |
Problem 5
Two months ago, the price, in dollars, of a cell phone was \(c\).
- Last month, the price of the phone increased by 10%. Write an expression for the price of the phone last month.
- This month, the price of the phone decreased by 10%. Write an expression for the price of the phone this month.
- Is the price of the phone this month the same as it was two months ago? Explain your reasoning.