Ahmad and Dante wash cars in the neighborhood and charge $5 per car. Last year, they worked very hard and were able to wash 450 cars in the year. They have done a little market research and found that for every $0.20 that the price is reduced, they can get 50 more customers.

2. a. Model the relationship between the number of $0.20 price decreases and the amount of money earned using an equation, table, and graph.

The model for the relationship between the number of $0.20 price decrease, x, to the amount of money earned, E, is presented by the following equation;

E = -10•x² + 160•x + 2250

Please find attached the values of the model in the attached table and graph of the model.

How can the equation of the model be found?

Let x represent the number of $0.20 price decrease and let E represent the total amount earned (the revenue), we have the following revenue equation;

E = Amount earned = Number of cars washed × The price to wash one car

However;

When the number of cars washed = (450 + 50•x)

The price to wash one car = (5 - 0.2•x)

Therefore;

E = (450 + 50•x) × (5 - 0.2•x)

Which gives the following equation;

E = -10•x² + 160•x + 2250

Please find the attached table and graph created using the above equation for the amount earned per year.

From the graph, the maximum amount is earned when the price is reduced by $0.20, eight times to give;

Optimum price = 5 - 0.2×8 = 3.4

The optimum price = $3.4

The number of cars washed at the optimum price = 850

The maximum amount earned = $2890

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