The cost(in$) to buy a given amount of gas(in gallons) plus a wash for
a Dodge Durango XLT is given by the following table.
       Gas(Gallons)        Cost($)-includes wash
           5                      14.7
          10                      22.65
          15                      30.60
          20                      38.55
          25                      46.50
          12.5
          27.5
a)Find the model c(x),which gives the total cost(in$)to buy x gallons
of gas and a wash. Remember Units!

b)Find the derivative model. Remember Units!

c)Use the model from part a) to complete the table.

d)From the model what is the price per gallon of the gas?

e)How much does the wash cost, by itself?

Hi and thanks for the question.

a) The model that gives the total cost in dollars for buying x gallons
of gas and a wash is:

Total Cost = (# of gallons)* $1.59 + $6.75

I got this by taking the following two equations from the chart:

14.7 = (cost per gallon)* 5 + (cost of car wash)

and 

22.65 = (cost per gallon)* 10 + (cost of car wash)

I then isolated (cost per gallon) in the first equation and used
substitution to find the value of (cost of car wash).  I then used the
value of (cost of car wash) to solve for (cost per gallon). Let me
know if you want further clarification on this procedure.


b) The derivative model is going to be just y'=$1.59 since the
derivative of 1.59x+6.75 is 1.59.

c) Let us use the model to complete the table:

For 12.5 gal:

total cost = $1.59 * 12.5 gal + $6.75

total cost = $26.625

For 27.5 gal:

total cost = $1.59 * 27.5 gal + $6.75

total cost = $50.475

d. The price per gallon of gas is the slope of the y=mx+b equation or
$1.59.

e. The price of the car wash is the y intercept of the y=mx+b equation
or $6.75. One way to see this is that if you calculated how much the
cost would be for getting zero gallons of gas using the model, the
result would be the y intercept value or $6.75

Search Terms
linear equations

Please ask for clarification of any part before rating this answer. 

Best of luck!

-blinkwilliams

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Request for Answer Clarification by fredewq-ga on 04 Dec 2002 21:56 PST

Would you please claify the procedure for me to isolate cost per
gallon and substitution to find the value of the cost of car wash.

Thanks

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Clarification of Answer by blinkwilliams-ga on 04 Dec 2002 22:34 PST

Hi, let me see if I answer your clarification. We know that for 5
gallons plus a car wash the cost is $14.70.  We also know that for 10
gallons plus a car wash the cost is $22.65. This gives us the
following two equations:

1st equation: $14.70=(cost per gallon)*5 gallons + (cost of car wash)
2nd equation: $22.65=(cost per gallon)*10 gallons + (cost of car wash)

Now let's solve the first equation for the (cost per gallon):

$14.70 - (cost of car wash) =(cost per gallon)*5 
($14.70 - (cost of car wash))/5 = (cost per gallon)

So now we can substitute ($14.70 - (cost of car wash))/5 for (cost per
gallon) in the second equation:

$22.65 = ($14.70 - (cost of car wash))/5  *10 + (cost of car wash)

This gives us:

$22.65 = $29.40 - (cost of car wash)

or (cost of car wash) = $6.75

Now that we know the cost of the car wash we can plug that value back
into the 1st equation to find the (cost per gallon):

$14.70=(cost per gallon)*5 + $6.75

Now we get: (cost per gallon) = $1.59

Now we know that the (cost per gallon) = $1.59 and (cost of car wash)
= $6.75

So we can use these two values to construct the model:

Total Cost = (# of gallons)* $1.59 + $6.75 

Further elaboration on this method of solving equations can be found
at:
http://www.ucl.ac.uk/Mathematics/geomath/rev/simnb/sim6.html 
 
Hope that helps! Let me know if further clarification is needed!

Best,

-blinkwilliams-ga