Question

for exercises 1 and 2, graph the data. then find the slope. explain what the slope represents.

1. envelopes the table shows the number of envelopes stuffed for various times.

time (min) 5 10 15 20
envelopes stuffed 30 60 90 120

1. measurement there are 3 feet for every yard
2. use the graph that shows the number of laps completed over time. find the slope of the line
3. which line is the steepest? explain using the slopes of lines l, m, and n

Explanation:

Step1: Recall slope formula

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.

Step2: Solve for problem 1

Let $x$ be time (min) and $y$ be envelopes stuffed. Taking two points, say $(5,30)$ and $(10,60)$. Then $m=\frac{60 - 30}{10 - 5}=\frac{30}{5}=6$. The slope represents the rate at which envelopes are stuffed per minute, i.e., 6 envelopes are stuffed per minute.

Step3: Solve for problem 2

Let $x$ be yards and $y$ be feet. Since there are 3 feet for every yard, if we take two points $(1,3)$ and $(2,6)$, then $m=\frac{6 - 3}{2 - 1}=3$. The slope represents the number of feet per yard, which is 3.

Step4: Solve for problem 3

From the graph of laps completed over time, taking two points, say $(10,5)$ and $(20,10)$. Then $m=\frac{10 - 5}{20 - 10}=\frac{5}{10}=0.5$. The slope represents the rate of laps completed per minute, i.e., 0.5 laps per minute.

Step5: Solve for problem 4

To determine the steepest line among $l$, $m$, and $n$, we find the absolute - value of their slopes. The line with the largest absolute - value of the slope is the steepest. Calculate the slopes of each line using two points on each line and then compare the absolute - values of the slopes.

Answer:

For problem 1: Slope is 6, representing 6 envelopes stuffed per minute.
For problem 2: Slope is 3, representing 3 feet per yard.
For problem 3: Slope is 0.5, representing 0.5 laps per minute.
For problem 4: Calculate slopes of $l$, $m$, $n$ using two - point formula and compare absolute values to find the steepest line.