106 H1.05: Example 3 Alternative Method

Alternative method of solution for Example 3:

Example 3 question: For a certain type of letter sent by Federal Express, the charge is $8.50 for the first 8 ounces and $0.90 for each additional ounce (up to 16 ounces.)   How much will it cost to send a 12-ounce letter?

We saw here that it is awkward for the value to be so far from any useful values for the variables. This suggests that we re-define the variables to be more useful.   

1-2. Define the variables and their units and possible values. List the points.

Let x = number of ounces above 8. So the possible values for x are 0 to 8 (which corresponds to the letter weighing 8 ounces to 16 ounces.)

Let y = cost. The possible values for y are $8.50 and larger.

A letter weighing 8 ounces costs $8.50. So x = 0. The point is (0, 8.50)

A letter weighing 9 ounces costs $8.50+0.90 = $9.40.   So x = 1. The point is (1, 9.40)

3. Is a linear model appropriate?

Yes, because the cost increases by the same amount for each additional ounce.

4-5. Find the slope and the formula.

Using the points above, we have

6. Interpret the y-intercept.

When x = 0, the value for y is 8.50. Since x = weight – 8 ounces, that means that when the letter weighs 8 ounces, the cost is $8.50.

Notice that defining the x-variable in this manner has made the y-intercept more meaningful in the problem.

7. Make the prediction.

If the letter weighs 12 ounces, then we use x = 12 – 8 = 4. So [latex]y=8.50+0.90\cdot4=8.50+3.60=12.10[/latex].

This agrees with the answer we found by the first method.

8. Graph and check whether you obtain the same results.

By hand, plot these points, draw the line, extend it to the limits for the variables, and use it to estimate the answer to the question.

On the graph, look up a weight of 12 ounces, which is
x = 12 – 8 = 4.

The cost for x = 4 is a little larger than $12, which is consistent with the value we obtained from the formula.