Lesson 9
Linear Models
Problem 1
![Four images of the moon, each with a larger crescent illuminated.](https://cms-im.s3.amazonaws.com/2f7SbvXKjJt5z1GVohRwPoPa?response-content-disposition=inline%3B%20filename%3D%228-8.5.C.PP_moons_clip_art.png%22%3B%20filename%2A%3DUTF-8%27%278-8.5.C.PP_moons_clip_art.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T202330Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=308868ab10fcad9271eec65634d3aa017b02191a13e4bd627ef194b997ae3941)
On the first day after the new moon, 2% of the Moon's surface is illuminated. On the second day, 6% is illuminated.
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Based on this information, predict the day on which the Moon’s surface is 50% illuminated and 100% illuminated.
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The Moon’s surface is 100% illuminated on day 14. Does this agree with the prediction you made?
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Is the percentage illumination of the Moon’s surface a linear function of the day?
Solution
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Problem 2
In science class, Jada uses a graduated cylinder with water in it to measure the volume of some marbles. After dropping in 4 marbles so they are all under water, the water in the cylinder is at a height of 10 milliliters. After dropping in 6 marbles so they are all under water, the water in the cylinder is at a height of 11 milliliters.
- What is the volume of 1 marble?
- How much water was in the cylinder before any marbles were dropped in?
- What should be the height of the water after 13 marbles are dropped in?
- Is the relationship between volume of water and number of marbles a linear relationship? If so, what does the slope of a line representing this relationship mean? If not, explain your reasoning.
Solution
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Problem 3
Solve each of these equations. Explain or show your reasoning.
\(2(3x+2)=2x+28\)
\(5y+13=\text-43-3y\)
\(4(2a+2)=8(2-3a)\)
Solution
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(From Unit 4, Lesson 13.)Problem 4
For a certain city, the high temperatures (in degrees Celsius) are plotted against the number of days after the new year.
Based on this information, is the high temperature in this city a linear function of the number of days after the new year?
![Coordinate plane, horizontal, negative 1 to 14 by ones, vertical, negative 4 to 16 by twos. Three points are drawn but not labeled, (1 comma negative 3), (3 comma 1), (10 comma 15).](https://cms-im.s3.amazonaws.com/AGVHYa2CENY6bkc6zjJncBKJ?response-content-disposition=inline%3B%20filename%3D%228-8.5.C8.PP.Image.103.png%22%3B%20filename%2A%3DUTF-8%27%278-8.5.C8.PP.Image.103.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T202330Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ab99ff45b130e1b092440328b40078bf337799309eb3138d32db2d7a12181a37)
Solution
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Problem 5
The school designed their vegetable garden to have a perimeter of 32 feet with the length measuring two feet more than twice the width.
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Using \(\ell\) to represent the length of the garden and \(w\) to represent its width, write and solve a system of equations that describes this situation.
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What are the dimensions of the garden?
Solution
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(From Unit 5, Lesson 16.)