Question

the team also wants to look at how much money they will make by selling caps. a table of data is shown. use this table to answer the following questions.
caps sold
number of caps | amount collected in dollars
0 | 0.00
1 | 10.00
2 | 20.00
3 | 30.00
4 | 40.00
5 | 50.00

a. graph the data on the coordinate grid drawn in question 2.
b. find the first differences in the table and draw the slope triangles on the graph you constructed.
c. what are the vertical and horizontal lengths of the slope triangles for the “caps sold” line?
d. how do the vertical and horizontal lengths of the slope triangles connect to the problem situation?
e. how do the vertical and horizontal lengths of the slope triangles connect to the first differences of the table?

1. how are the slope triangles in the graph, the first differences in the tables, the rates of change in the problem situation, and the equations of the best lines connected?

1. using the terms slope and y - intercept, explain how the graph shows that the amount of money collected will eventually be more than the amount paid. express your answer as a complete sentence.

Explanation:

Step1: Analyze the data table

The table has two columns: "number of caps" (let's call this \( x \)) and "Amount collected (in dollars)" (let's call this \( y \)). The values are:
- When \( x = 0 \), \( y = 0.00 \)
- \( x = 1 \), \( y = 10.00 \)
- \( x = 2 \), \( y = 20.00 \)
- \( x = 3 \), \( y = 30.00 \)
- \( x = 4 \), \( y = 40.00 \)
- \( x = 5 \), \( y = 50.00 \)

Step2: Find the first differences (for part b)

First differences are the differences between consecutive \( y \)-values.
- Between \( x = 0 \) and \( x = 1 \): \( 10.00 - 0.00 = 10.00 \)
- Between \( x = 1 \) and \( x = 2 \): \( 20.00 - 10.00 = 10.00 \)
- Between \( x = 2 \) and \( x = 3 \): \( 30.00 - 20.00 = 10.00 \)
- Between \( x = 3 \) and \( x = 4 \): \( 40.00 - 30.00 = 10.00 \)
- Between \( x = 4 \) and \( x = 5 \): \( 50.00 - 40.00 = 10.00 \)

So the first difference is constant at \( 10.00 \) dollars per cap.

Step3: Slope triangles (part b and e)

The slope of a line is \( \frac{\text{vertical change}}{\text{horizontal change}} \). For the "Caps sold" line, the horizontal change (change in \( x \)) between consecutive points is \( 1 \) (since the number of caps increases by 1 each time), and the vertical change (change in \( y \)) is \( 10 \) (the first difference). So the slope triangles will have a horizontal length of \( 1 \) (representing 1 cap) and a vertical length of \( 10 \) (representing $10$ collected). This matches the first differences (which are $10$) because the first difference is the vertical change for a horizontal change of \( 1 \), which is the slope (vertical/horizontal) of the slope triangles.

Step4: Connection between slope triangles, first differences, rate of change, and best - fit line (part f)

- Slope triangles: The slope of the slope triangles is \( \frac{\text{vertical length}}{\text{horizontal length}} \). Here, vertical length is the first difference ($10$) and horizontal length is \( 1 \) (change in number of caps).
- First differences: The first difference is the vertical change in the \( y \)-values for a horizontal change of \( 1 \) in \( x \)-values, which is the rate of change.
- Rate of change: In the problem situation, the rate of change is the amount of money collected per cap, which is \( \frac{10}{1}=10 \) dollars per cap.
- Best - fit line: For a linear relationship (which this is, since first differences are constant), the best - fit line will have a slope equal to the rate of change (first difference per unit horizontal change), and the slope triangles will have slopes equal to the slope of the best - fit line.

Step5: Explanation for part g (using slope and y - intercept)

The equation of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. From the table, when \( x = 0 \) (no caps sold), \( y = 0 \), so the \( y \)-intercept \( b = 0 \). The slope \( m \) is the rate of change, which we found to be \( 10 \) (since for each increase of \( 1 \) in \( x \) (number of caps), \( y \) (amount collected) increases by \( 10 \)). So the equation is \( y = 10x+0 \) or \( y = 10x \). Since the slope \( m = 10>0 \), as \( x \) (number of caps sold) increases, \( y \) (amount collected) will increase linearly. So the graph (a line with positive slope and \( y \)-intercept \( 0 \)) shows that as the number of caps sold (\( x \)) increases, the amount of money collected (\( y \)) will increase, and since the slope is positive and the \( y \)-intercept is \( 0 \), the amount collected will be more than the amount paid (assuming the amount paid is a fixed cost or zero - in this case, when \( x = 0 \), amount collected is \( 0 \), and as \( x \) increases, \( y = 10x \) will be greater than \( 0 \) for \( x>0 \)).

Answer:

Part b:

The first differences are all \( 10.00 \) dollars. The slope triangles on the graph will have a vertical length of \( 10 \) (the first difference) and a horizontal length of \( 1 \) (change in number of caps).

Part c:

The vertical length of the slope triangles represents the change in the amount of money collected (in dollars), and the horizontal length represents the change in the number of caps sold (by 1 cap).

Part d:

The vertical length of the slope triangles is the increase in the amount of money collected when the number of caps sold (horizontal length) increases by 1. For example, a horizontal length of 1 (1 more cap sold) corresponds to a vertical length of 10 (10 more dollars collected), which is the rate of money collected per cap.

Part e:

The vertical length of the slope triangles is equal to the first difference of the table (since first difference is the change in \( y \) for a change of 1 in \( x \)), and the horizontal length of the slope triangles is 1 (the change in \( x \) between consecutive data points), so \( \text{slope}=\frac{\text{vertical length}}{\text{horizontal length}}=\text{first difference} \).

Part f:

The slope of the slope triangles, the first difference, and the rate of change (money per cap) are all equal (to 10). The best - fit line for this linear data will have a slope equal to this common value, and the slope triangles will have slopes equal to the slope of the best - fit line.

Part g:

The graph of the "Caps sold" relationship is a line with a \( y \)-intercept of \( 0 \) (when no caps are sold, no money is collected) and a slope of \( 10 \) (for each cap sold, $10$ is collected). Since the slope is positive, as the number of caps sold (\( x \)) increases, the amount of money collected (\( y = 10x \)) will increase. So, as more caps are sold (larger \( x \)), the amount of money collected will be more than the amount paid (assuming the amount paid is a fixed cost or zero, and for \( x>0 \), \( 10x>0 \) when \( x > 0\)).