Word Problem

Question

You want to decide between buying a 2015 Toyota Camry and a 2015 Toyota Camry Hybrid. The hybrid costs more upfront, but saves you money over the life of the vehicle because of gas. Exactly how long will it take to pay for the extra initial cost so you are saving money each year?


Some information:
The Hybrid costs $26,790 and the non-hybrid costs $22,970.

The cost of gas per year for driving the hybrid is $477.94. (This is found by using the following information: MPG rating for hybrid is 43 city and 35 hwy. Average person in Washington State drives 8,452 miles per year. We will assume of this, that we drive 75% of these miles in the city and 25% of the miles on the hwy, This is equivalent to driving 6339 miles in the city and 2113 miles on the hwy per year. If we divide the number of miles by the MPG rating we will get the number of gallons we use each year in the city and on the hwy. This is 147.4 gallons for driving in the city and 60.4 gallons for driving on the hwy. If we make the assumption that the average cost of a gallon of gas is $2.30, then we can calculate that it will cost us $339.02 for driving each year in the city and $138.92 for driving each year on the hwy. That is a total cost of $477.94 per year for the cost of gas each year….)

The cost of gas per year for driving the non-hybrid is $722.11. (This is found by using the following information: MPG rating for hybrid is 25 city and 35 hwy. Average person in Washington State drives 8,452 miles per year. We will assume of this, that we drive 75% of these miles in the city and 25% of the miles on the hwy, This is equivalent to driving 6339 miles in the city and 2113 miles on the hwy per year. If we divide the number of miles by the MPG rating we will get the number of gallons we use each year in the city and on the hwy. This is 253.56 gallons for driving in the city and 60.4 gallons for driving on the hwy. If we make the assumption that the average cost of a gallon of gas is $2.30, then we can calculate that it will cost us $583.19 for driving each year in the city and $138.92 for driving each year on the hwy. That is a total cost of $722.11per year for the cost of gas each year….)

For the scenario, you are to complete the following :


1. Identify an equation that represents the total cost (original price plus the cost of gas per year) of the hybrid based on the number of years driven.


2. Identify an equation that represents the total cost (original price plus the cost of gas per year) of the non-hybrid based on the number of years driven.


3. For each of the two equations, create a table of ordered pairs (at least 5 each) that would be solutions to the equation. (This means you will have two different tables…)


4. Construct a visual representation of the two equations on the same coordinate plane. Make sure to appropriately label each of the axes. Please identify each equation either by color or other identifying label.

5. For each equation, interpret the following (which means, give me the value and then explain what that means in context to the situation that is being discussed):
1. y-intercept
2. Slope


6. Find the point of intersection of the two lines. (You may do this by reading it off the graph, substitution, or elimination.)


7. Interpret this point in context within this problem.
(In words, describe the answer to the original question posed in the problem.)

Answer 1

1) y = 26790 + 477.94X , the hyrbid

2) y = 22970 + 722.11X , the non-hybrid

3) table created in ms excel

year hybrid non-hybrid
0      26790 22970
1 27267.94 23692.11
2 27745.88 24414.22
3 28223.82 25136.33
4 28701.76 25858.44
5 29179.7 26580.55
6 29657.64 27302.66
7 30135.58 28024.77
8 30613.52 28746.88
9 31091.46 29468.99
10 31569.4 30191.1

4) graph is left for you

5) For the hybrid model, the slope is 477.94 and the intercept is 26790.
This means the cost of the vehicle increases 477.94 annually due to gas
beginning with 26790.

For the non-hybrid model, the slope is 722.11 and the intercept is 22970.
This means the cost of the vehicle increase 722.11 annually due to gas
beginning with 22970.

For ANY line in general, the y-intercept measures where the line crosses the
y-axis. The slope measures the change in y-value divided by the change in x-value
for any two points on the line. This ratio, proportion, or fraction representing
the slope M is the same REGARDLESS of which two points on the line are selected.

6) y = 26790 + 477.94X = 22970 + 722.11X <-- sets them equal and solves for X

Moving everything to the right side:
0 = 22970 + 722.11X - 26790 - 477.94X

0 = 244.17X - 3820

244.17X = 3820

X = 15.644

Check: 26790 + 477.94 * 15.644 = 34267 rounded to the nearest dollar
22970 + 722.11 * 15.644 = 34267 rounded to the nearest dollar

7) It will take 15 years and almost 8 months for the cost of the
non-hybrid to catch up to the expense of the hybrid.

The difference in the prices of the cars is $3820. You are saving
244.17 per year on gas with the hybrid. These figures appear in
part 6, verifying the answer that the savings in gas will be gone
after 15 years and almost 8 months.

So if you plan to keep the car for 15 years, you will have
saved $3662.15 in gas.