Question

question 10
distance traveled (feet)
number of revolutions
(a) how many revolutions does mary make per foot of distance traveled?
□ revolution(s)
(b) what is the slope of the graph?
□

Explanation:

Step1: Analyze the graph for part (a)

From the graph, when distance (y) is, say, 24 feet, number of revolutions (x) is 6. To find revolutions per foot, we calculate \( \frac{\text{revolutions}}{\text{distance}}=\frac{6}{24}=\frac{1}{4} \)? Wait, no, wait. Wait, the x - axis is number of revolutions, y - axis is distance. Wait, the question is revolutions per foot. So it's \( \frac{\text{revolutions}}{\text{foot}} \). Let's take a point. Let's see, when x (revolutions) is 4, y (distance) is 16? Wait no, looking at the graph: when x = 4, y (distance) is 16? Wait no, the y - axis is distance (feet), x - axis is number of revolutions. Wait, the line passes through (4, 16)? Wait no, when x = 2, y = 8? Wait, when x = 4, y = 16? Wait, no, let's check the coordinates. The origin is (0,0). Let's take x = 4 (revolutions), y = 16 (feet)? Wait, no, the y - axis at x = 4: looking at the graph, when x = 4, y is 16? Wait, no, the grid: each x - unit (revolution) and y - unit (foot). Wait, maybe a better point: when x = 8 (revolutions), y = 32 (feet)? Wait, no, the graph's y - axis at x = 8: the blue line goes up. Wait, actually, the slope of the graph (distance vs revolutions) is \( \frac{\text{distance}}{\text{revolution}} \), which is speed per revolution. But the question (a) is revolutions per foot, which is the reciprocal of the slope. Let's find two points. Let's take (0,0) and (4, 16). Wait, no, when x = 4 (revolutions), y = 16 (feet). So slope (distance per revolution) is \( \frac{16 - 0}{4 - 0}=4 \) feet per revolution. Then revolutions per foot is \( \frac{1}{4}=0.25 \). Wait, let's confirm with another point. (8, 32): slope is \( \frac{32}{8}=4 \) feet per revolution. So revolutions per foot is \( \frac{1}{4}=0.25 \), which is \( \frac{1}{4} \) or 0.25.

Step2: For part (a)

Revolutions per foot: \( \frac{\text{revolutions}}{\text{foot}} \). Using the slope of the graph (distance vs revolutions) is \( m=\frac{\text{distance}}{\text{revolution}} \). So revolutions per foot is \( \frac{1}{m} \). From the graph, let's take two points (x1, y1)=(0,0) and (x2, y2)=(4, 16) (wait, no, x is revolutions, y is distance. So (x = 4 revolutions, y = 16 feet). So slope \( m=\frac{y2 - y1}{x2 - x1}=\frac{16 - 0}{4 - 0}=4 \) feet per revolution. Then revolutions per foot is \( \frac{1}{4}=0.25 \) or \( \frac{1}{4} \).

Step3: For part (b)

The slope of the graph (distance vs revolutions) is \( \frac{\text{distance}}{\text{revolution}} \). Using the same points (0,0) and (4, 16), slope \( m=\frac{16 - 0}{4 - 0}=4 \). Or (8, 32): \( \frac{32 - 0}{8 - 0}=4 \). So the slope is 4.

Answer:

(a) \( \frac{1}{4} \) (or 0.25) revolutions per foot.
(b) The slope is 4.

(For part (a) box: \( \frac{1}{4} \) or 0.25; part (b) box: 4)