Diesel or Gasoline Engine?
Date: 10/16/96 at 23:24:59 From: Nick Fuller Subject: Diesel or Gasoline Engine? A new car may be purchased with either a gasoline or a diesel engine. The diesel engine gets 32 mpg. The gasoline engine gets 29 mpg. Gasoline cost $1.16 per gallon and diesel costs $1.04 per gallon. The diesel engine costs $427.65 more than the gasoline engine. How far must you drive a car with a diesel engine before the savings in fuel pay for the higher price of the engine? I've have came up with a couple of answers but don't think they are correct. I was wondering if you could help me out with something to check my answer with.
Date: 10/17/96 at 9:50:36 From: Doctor Keith Subject: Re: Diesel or Gasoline Engine? Hi Nick, You've got a nice problem. When facing a word problem it is always a good idea to try to get a feel for what is going on and what you think you should get as an answer. To help us, let's make a table of what we know: gas car diesel car miles per gallon(mpg) 29 32 price of fuel $1.16 $1.04 additional cost $0 $427.65 Notice a few things: 1) the diesel car will go farther on one gallon of fuel. 2) the diesel fuel costs less. 3) the diesel car costs more. So we know the diesel car costs more at first but as you drive it will cost less per mile than a gas car due to the mpg and the fuel cost. Thus at some point we expect the diesel car to be cheaper, counting the fuel savings. Since the fuel and the mpg are not that different and the cost of the engine is much more than the difference in the costs, we expect it to take a long time to get our money back. Why bother doing this? Well, first off it helps organize our thoughts. Second, it checks the problem and our comprehension of the problem. Third, we have some idea about what we should get (a big number). So it is a good idea to take some time to check before starting a problem. Okay, we are ready to start the problem. First we are looking for the miles we must drive until we pay off the $427.65 for the diesel engine. Let's call the number of miles we will have to drive x. Now we want to write expressions for how much it will cost for the gas and diesel cars to drive x miles. We will do this in two steps so you get the feel of it: Step 1: get the gallons of fuel we will use in x miles For the gas car: (gallons) x (x miles)/(29 miles per gallon)= (x miles) x ---------- = --- gallons (29 miles) 29 For the diesel car: (gallons) x (x miles)/(32 miles per gallon)= (x miles) x ---------- = --- gallons (32 miles) 32 Step 2: get the cost for that much fuel gas: ((x/29) gallons)(1.16 dollars per gallon)= x(1.16/29) dollars diesel: ((x/32) gallons)(1.04 dollars per gallon)= x(1.04/32) dollars Now that we know how much it will cost in fuel to drive both x miles, we need to make one final observation. When we defined x we said it was the miles we had to drive so that we paid off the additional cost of a diesel engine. Thus we know that the cost of driving the gas car x miles is equal to the cost of driving the diesel car plus the cost of the diesel engine. In a math equation this is: x(1.16/29) = x(1.04/32) + 427.65 So all we need to do is solve for x. x(0.04) = x(0.0325) + 427.65 x(0.04) - x(0.0325) = 427.65 x((0.04) - (0.0325)) = 427.65 x(0.0075) = 427.65 x = 427.65/0.0075 = 57,020 miles That is a lot of driving. Just as a frame of reference the average car gets driven 12,000 miles a year. Thus it will take us about four years and nine months of average driving to recover the initial investment! That is quite a while in my book. Hope this helps. -Doctor Keith, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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