Question

I've been stuck on this problem for a while now so if you could help that'd be great! Thanks

An automaker produces a car that can travel 40 miles on its charged battery before it begins to use gas. Then the car travels 50 miles for each gallon of gas used.

A) Represent the relationship between the amount of gas used and the distance traveled using a table and an equation.

B) Is the total distance traveled a function of the amount of gas used? What are the independent and dependent variables? Explain

Answer 1

Lets call the total gallons of gas used 'g' and the total distance traveled 'd'.

From the first sentence we know that the car can travel 40miles without using any gas, i.e. d=40 when g=0.

We also know that after 40miles, the car will travel 50miles for every 1 gallon of gas, i.e when g=0+1, d=40+50; when g=0+2, d=40+(50+50).

Using this information we can start to build our table:

g | d
-------
0 | 40
1 | 90
2 | 140
etc.

We could continue this table as far as we wanted to get whatever values we need. An easier way is to convert the information to a function.

Looking at the information we have we can see that we are only doing addition and multiplication; there are no squared values, logarithms etc. This tells us that our function is linear. it will have the general form: \[y=mx+b\]
Or, using our notation: \[d=gm+b\]

In order to find 'm' and 'b' we can plug in some the values from our table:

40 = 0m+b.

Solving shows us that b=40.

Similarly:

\[140 = 2m +b\]

But we already know b=40, so:

\[140 = 2m + 40\]

Solving tells us that m=50.

Our final function is then:

\[d=50g+40\]


Part 2 asks if distance is a function of gallons of gas. We have shown that yes, \[d=f(g)\]

specifically we have shown that \[d=50g+40\]

We can also see that since'd' depends on 'g', 'g' is the independent variable and 'd' is the dependent variable.