# The Sequel

Having solved the problem presented on the previous page, consider Act 1 of the sequel:

After this clip, students should deduce the following:

• We need to devise a model by which we can find the total cost of purchasing a given amount of fruit.
• We need to know the rate of sales tax in the transaction.

Upon student request to know the sales tax, we view Act II below:

For this problem, assume that the purchase is made within the Huntsville City limits.

Now, we have all of the information we need to formulate a model by which we can determine the total cost of purchasing a given amount of fruit. In the first problem, we determined that the unit price of an apple is \$0.93/apple and the unit price of a pear is \$0.75/pear, so we recognize that there is a linear relationship between how many pieces of fruit are purchased and the total cost of the purchase. Thus, we can model this relationship with two linear equations. If we choose to let x represent the number of fruit purchased and y represent the total cost of the purchase, which piece of information represents the slope of this linear relationship? Using the slope sliders in the Desmos graph below, graph the two linear equations, letting the red line represent a purchase of apples and the green line represent a purchase of pears.

*Note: The slope in each equation has a coefficient of 1.09 in order to account for the 9% sales tax applied to the final transaction.

The slopes of our linear equations relate the number of fruit purchased to the total cost. It makes sense that 0.93 is the slope of the apple equation while 0.75 is the slope of the pear equation, since these are the respective unit prices of the apples and pears. The Desmos graph below shows these two equations graphed correctly.

In the equation of the red line, x=the number of apples and the slope n= \$0.93/apple. In the equation of the green line, x=the number of pears and the slope m= \$0.75/pear. In both equations, y= the cost of the purchase and the coefficient of 1.09 accounts for sales tax.

Using this graph, we can answer the question posed in Act I of this problem: Does the total cost shown on the register align with our expectations of what the cost of three pieces of fruit should be, given the information we have?

On our graph, we find the y-values on each line that correspond to an x-value of x=3. By doing this, we see that the expected cost of three apples is approximately \$3.00, while the expected cost of three pears is approximately \$2.50. Based on this data, there is definitely a discrepancy between our calculations and the actual cost of the items. This provides an opportunity for us to reflect on potential causes of the discrepancy and methods that could improve the model. Some examples of such analysis include

• The scale at the checkout may have been calibrated differently than the scale in the produce section, since the checkout scale weighed 3 apples at 1.53 lbs (24.48 oz) and 3 pears at 0.92 lbs (14.72 oz).
• Some or all of the apples actually weighed more than 6 oz. and some or all of the pears actually weighed more than 4 oz.
• For a more accurate weight of an individual apple or pear, we could take a sample of ten pieces of each type of fruit, weighing each piece and using the mean weight from the samples.

Verifying our conclusion, we are rewarded with Act III:

Even though the graph of the equations revealed a discrepancy between the expected price for three pieces of fruit and the actual price, tracing the cause of the discrepancy to variance in the weight of the fruit serves to validate our linear model. By taking the weight of the fruit as shown on the cash register, converting it to ounces, then dividing by 6 oz./apple and 4 oz/pear, we see that our linear equations yield close approximations of the actual total cost.

Following any good narrative and sequel comes the review! Continue to the next page to see how this problem can be integrated into the classroom.