Question

mai is riding her bike. the number of revolutions (turns) her wheels make varies directly with the distance she travels. see the graph below.
(a) how many revolutions does mai make per foot of distance traveled?
revolution(s)
(b) what is the slope of the graph?

Explanation:

Step1: Analyze the direct variation

Since the number of revolutions \( y \) varies directly with distance \( x \), the relationship is \( y = kx \), where \( k \) is the constant of proportionality (revolutions per foot) and also the slope of the line.

Step2: Find two points on the graph

From the graph, when \( x = 30 \) feet, \( y = 6 \) revolutions (estimating from the line's path).

Step3: Calculate the slope (revolutions per foot)

The slope \( k=\frac{y}{x}=\frac{6}{30} = 0.2\) (or \(\frac{1}{5}\)). This is also the revolutions per foot.

Part (a)

Step1: Identify the formula for direct variation

For direct variation \( y = kx \), \( k=\frac{y}{x} \) gives revolutions per foot.

Step2: Use a point from the graph

Take \( x = 30 \), \( y = 6 \) (from the graph's grid). Then \( k=\frac{6}{30}=\frac{1}{5}=0.2 \).

Part (b)

Step1: Recall slope formula for a line

The slope \( m \) of a line through \((x_1,y_1)\) and \((x_2,y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Use the origin \((0,0)\) and \((30,6)\)

\( m=\frac{6 - 0}{30 - 0}=\frac{6}{30}=0.2 \) (or \(\frac{1}{5}\)).

Answer:

(a) \( \boldsymbol{0.2} \) (or \( \boldsymbol{\frac{1}{5}} \)) revolution(s) per foot
(b) \( \boldsymbol{0.2} \) (or \( \boldsymbol{\frac{1}{5}} \))