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Story problem

A company charges a fixed fee for renting a moving van, in addition to charging per mile driven. If a person rents a moving van and drives it 100 miles the charge is $35. If a person rent a moving van and drives it 160 miles the charge is $44. 
A) find a linear function f(x)= mx+b whose input is the number of miles driven and whose output is the corresponding charge for renting the moving van. 
B) use the function from part a to determine the cost of renting a moving can and driving it 300 miles.
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1 Answer

Hi, Tiphany!  Let's see if this helps:
 
I'm going to assume you understand what a function is, and that you are being asked to show your answer as a slop-intercept form of a straight line.  So, let's get right to the heart of it.
 
First, we can figure out the slope (m) by imagining two ordered pairs, both being the number of miles driven (x), and the costs (y).  Thus, we have two points; (100,35) and (160,44).  Using the formula (y2-y1)/(x2-x1), we can plug in our points to garner the following:  (44-35)/(160-100), which comes out to 9/60, or reduced further, 3/20.
 
Now we have f(x) = 3/20x + b.
 
b is easy to figure out from here.  If we plug in our first mileage (x), we end up with f(x) = 3/20(100) + b, which is f(x) = 15 + b.  Because we know the cost for 100 miles is $35, we can deduce b must be 20, as 15 + 20 = 35.
 
But, let's not take that as gospel yet.  Let's test b = 20 when we have 160 miles, to make sure it is correct.  So, if we solve f(x) = 3/20(160) + 15, do we get 44?  f(x) = 24 + 20; f(x) = 44.  Yes we do.
 
I hope this helped!

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