Section B Section B Checkpoint

Problem 1

The relationship between the circumference of a circle and its radius is represented by this graph:

Graph of the relationship between the circumference of a circle and its radius. Horizontal, radius in centimeters, vertical circumference in centimeters.

The perimeter in centimeters, $P$ , of a rectangle whose length is twice the size of its width in centimeters, $w$ , is given by the equation $P=6w$ .

Compare the outputs of the two functions when the inputs for each function is 4.
For the relationship between $P$ and $w$ , name the independent and dependent variables.
Han says that the circumference is increasing faster than the perimeter. What do you think he means by that?

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Solution

Sample response: When the radius of the circle is 4 cm, the circumference is $8\pi$ cm. When the width of the rectangle is 4 cm, the perimeter is 24 cm.
Sample response: For the equation $P=6w$ , we can input $w$ to calculate the value of $P$ , the output. So $w$ is the independent variable, and $P$ is the dependent variable.
Sample response: The rate of change for the circumference is $2\pi$ cm (or about 6.28 cm) per 1 cm increase in the input, while the rate of change for the perimeter of the rectangle is 6 cm per 1 cm increase in input. Since the output changes by more for each increase of the input by 1 cm, the circumference is increasing faster.

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Sample Response

Sample response: When the radius of the circle is 4 cm, the circumference is $8\pi$ cm. When the width of the rectangle is 4 cm, the perimeter is 24 cm.
Sample response: For the equation $P=6w$ , we can input $w$ to calculate the value of $P$ , the output. So $w$ is the independent variable, and $P$ is the dependent variable.
Sample response: The rate of change for the circumference is $2\pi$ cm (or about 6.28 cm) per 1 cm increase in the input, while the rate of change for the perimeter of the rectangle is 6 cm per 1 cm increase in input. Since the output changes by more for each increase of the input by 1 cm, the circumference is increasing faster.

Problem 2

The graph shows Tyler’s distance from school as a function of time since school ended.

Coordinate plane, horizontal, time in hours, 0 to 5. Vertical, distance from school in miles, 0 to 2.

Clare walks home right after school. She stays home for an hour, then walks back to school to go to the volleyball game. After the game, she returns home.

Sketch a graph of Clare’s story.
Which quantity is a function of which? Explain your reasoning.
Based on your graph, is Clare’s house closer to school than Tyler’s house? Explain how you know.

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Solution

Sample response:

Coordinate plane, horizontal, time in hours, 0 to 5. Vertical, distance from school in miles, 0 to 2. Tyler's distance from school is horizontal from 0 comma 0 to 1 comma 0, then linearly upward to 1.5 comma 0.5, then horizontal to 5 comma 0.5. Clare's distance from school is linearly upward from 0 comma 0 to 0.5 comma 0.75, horizontal to 1.5 comma 0.75, then linearly downward to 2 comma 0, then horizontal to 3.5 comma 0, then linearly upward to 4 comma 0.75, then horizontal to 5 comma 0.75.
The distance from the school is a function of time since school ended. Sample reasoning: Clare is at the school at different times, so time cannot depend on distance from the school. For each time, there is one and only one value of distance, so distance must depend on time.
Sample response: No, Clare’s house is farther from school than Tyler’s because the graphs show that Tyler’s house is 0.5 miles from the school and Clare’s house is 0.75 miles from the school.

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Sample Response

Sample response:

Coordinate plane, horizontal, time in hours, 0 to 5. Vertical, distance from school in miles, 0 to 2. Tyler's distance from school is horizontal from 0 comma 0 to 1 comma 0, then linearly upward to 1.5 comma 0.5, then horizontal to 5 comma 0.5. Clare's distance from school is linearly upward from 0 comma 0 to 0.5 comma 0.75, horizontal to 1.5 comma 0.75, then linearly downward to 2 comma 0, then horizontal to 3.5 comma 0, then linearly upward to 4 comma 0.75, then horizontal to 5 comma 0.75.
The distance from the school is a function of time since school ended. Sample reasoning: Clare is at the school at different times, so time cannot depend on distance from the school. For each time, there is one and only one value of distance, so distance must depend on time.
Sample response: No, Clare’s house is farther from school than Tyler’s because the graphs show that Tyler’s house is 0.5 miles from the school and Clare’s house is 0.75 miles from the school.