Textbook answer to problem puzzling

[csmyth3025]

I'm tediously working my way through the first chapter of "Basic Technical Mathematics with Calculus", Seventh Edition, by Allyn J. Washington. This is strictly self-study on my part so I don't have an instructor to clarify things for me.

 

The word problem given is:

"Two gasoline distributors A and B are 228 miles apart on Interstate 80. A charges $0.85 per gallon and B charges $0.80 per gallon. Each charges $0.05 per mile for delivery. Where on Interstate 80 is the cost to the customer the same?"

 

I first designated the cost of delivered gasoline from A = $0.90/gal*mi and the cost of delivered gasoline from B = $0.85/gal*mi.

 

I then designated x = miles from A.

 

I then calculated as follows:

 

[math](0.90)(x) = (0.85)(228-x)[/math]

 

[math]0.90x = 193.8-0.85x[/math]

 

[math]0.90x+0.85x = 193.8[/math]

 

[math]1.75x = 193.8[/math]

 

[math]x = 110.743[/math] (miles from A)

 

[math]228-x = 117.257[/math] (miles from B)

 

If I use these mileages and multiply by the cost of delivered gas per gallon per mile I get: [math](0.90)(110.743) = 99.67[/math] for the total cost of gasoline from A.

 

Likewise, the total cost for delivered gas from B would be: [math](0.85)(117.257) = 99.67[/math]

 

This all seemed rather straightforward until I checked my answer against the answer for this exercise given in the back of the book. The given answer is 64 miles from A.

 

I've turned this problem around every way I can think of and there doesn't seem to be any way I can come up with 64 miles from A. What am I missing?

 

Chris

[imatfaal]

Unless I am mistaken the book is wrong (and so I think are you)

 

to test their answer

 

64 miles from A Price = 85 + 64*5 = 405

 

228-64 miles from B Price = 80 + 164*5 = 900

 

I would do it like this

 

85 +5x = 80 +5(228-x)

85 +5x = 80 +1140 - 5x

10x = 80 +1140 - 85

10x = 1135

x = 113.5

 

check

 

113.5 miles from A Price = 85 + 113.5*5 = 652.5

114.5 miles from P Price = 80 + 114.5*5 = 652.5

 

You are multiplying the fuel cost by the delivery distance in your method and that does not work. I will work out where the mistake in the book - or of our interpretation of the book and post

 

Got it. The delivery cost is 0.05 cents! Not 5 cents. In words the difference in the cost of the fuel must equal the difference in the distance times the delivery cost. 164m-64m is 100m and that difference in distance must equate to the 5c difference in price

 

64 miles from A Price = 85 + 64* 0.05 = 88.2

 

228-64 miles from B Price = 80 + 164*0.05 = 88.2

[csmyth3025]

You're absolutely right about my wrong calculation and also about the cost of delivery being $0.0005 per mile! The question (as written in the book) gives the cost of gasoline as $0.85 and $0.80 and the cost of delivery as 0.05 cents (using that symbol - I don't know how to make it on my keyboard).

 

This is a good lesson for me to read these problems more carefully and also to think through my approach to the solutions more logically.

 

Thanks

 

Chris

[Tres Juicy]

"Two gasoline distributors A and B are 228 miles apart on Interstate 80. A charges $0.85 per gallon and B charges $0.80 per gallon. Each charges $0.05 per mile for delivery. Where on Interstate 80 is the cost to the customer the same?"

 

 

 

surely the answer will vary depending on the size of the order?

 

I mean if I order 1 gallon from A at $0.85 and 1 gallon from B at $0.80 the difference in price is $0.05

 

So the point at which the price is the same from both companies is 114.5 miles from B and 113.5 miles from A

 

 

However, if I order 6 gallons the difference in price $0.30 so at the same distances the price from A including mileage would be $10.775 and the price from B is $10.525

 

Do you see what I mean?

 

 

[imatfaal]

"Two gasoline distributors A and B are 228 miles apart on Interstate 80. A charges $0.85 per gallon and B charges $0.80 per gallon. Each charges $0.05 per mile for delivery. Where on Interstate 80 is the cost to the customer the same?"

 

 

 

surely the answer will vary depending on the size of the order?

 

I mean if I order 1 gallon from A at $0.85 and 1 gallon from B at $0.80 the difference in price is $0.05

 

So the point at which the price is the same from both companies is 114.5 miles from B and 113.5 miles from A

 

 

However, if I order 6 gallons the difference in price $0.30 so at the same distances the price from A including mileage would be $10.775 and the price from B is $10.525

 

Do you see what I mean?

 

 

 

 

Yes but your delivery cost is related to amount you order as well. Using the flawed original delivery price

 

113.5 miles from A Price = 85*6 + 113.5*5*6 = (85 + 113.5*5)*6 = 652.5 * 6

114.5 miles from P Price = 80*6 + 114.5*5*6 = (80 + 114.5*5)*6 = 652.5 * 6

 

Still the same. Your argument would only work if the delivery cost were independent of the quantity - I think that could be a later question

[Tres Juicy]

Yes but your delivery cost is related to amount you order as well. Using the flawed original delivery price

 

113.5 miles from A Price = 85*6 + 113.5*5*6 = (85 + 113.5*5)*6 = 652.5 * 6

114.5 miles from P Price = 80*6 + 114.5*5*6 = (80 + 114.5*5)*6 = 652.5 * 6

 

Still the same. Your argument would only work if the delivery cost were independent of the quantity - I think that could be a later question

 

 

"your delivery cost is related to amount you order"

 

How? it doesn't say that - it says $0.05 per mile

[imatfaal]

"your delivery cost is related to amount you order"

 

How? it doesn't say that - it says $0.05 per mile

 

Youre quite right - it seems I had assumed that in order for the question to be answerable.

[Tres Juicy]

Youre quite right - it seems I had assumed that in order for the question to be answerable.

 

 

The question is broken, unless it tells you the amount of gallons ordered

[D H]

The word problem given is:

"Two gasoline distributors A and B are 228 miles apart on Interstate 80. A charges $0.85 per gallon and B charges $0.80 per gallon. Each charges $0.05 per mile for delivery. Where on Interstate 80 is the cost to the customer the same?"

As stated, the problem is unsolvable. You apparently have the wrong wording. The delivery cost should be $0.05 per gallon per mile.

[Daedalus]

"Two gasoline distributors A and B are 228 miles apart on Interstate 80. A charges $0.85 per gallon and B charges $0.80 per gallon. Each charges $0.05 per mile for delivery. Where on Interstate 80 is the cost to the customer the same?"

...

The given answer is 64 miles from A.

As everyone has stated, it would seem you are missing the quantity of gas being ordered. Let's write the problem as follows:

 

Equation for A:

 

[math]0.85 \times G + 0.05 \times M[/math]

 

Equation for B:

 

[math]0.80 \times G + 0.05 \times M[/math]

 

Where [math]G[/math] is the number of gallons ordered and [math]M[/math] is the number of miles to delivery. The answer in the book is 64 miles from A. So we will cheat and use this answer which gives us the following distances:

 

Distance from A = [math]64[/math]

Distance from B = [math]228-64=164[/math]

 

Now let's substitute those values back into the equations:

 

Equation for A:

 

[math]0.85 \times G + 0.05 \times 64 = 0.85 \times G + 3.20[/math]

 

Equation for B:

 

[math]0.80 \times G + 0.05 \times 164 = 0.80 \times G + 8.20[/math]

 

How many gallons of gas should be ordered to make the answer true? We need to set both equations equal to each other and solve for G:

 

[math]0.85 \times G + 3.20 = 0.80 \times G + 8.20[/math]

 

[math]G = 100[/math]

 

If we ordered 100 gallons of gas then the cost to the us would be the same from both A and B if we were located 64 miles from A and 164 miles from B:

 

Cost from A:

 

[math]0.85 \times 100 + 0.05 \times 64 = 88.20[/math]

 

Cost from B:

 

[math]0.80\times 100 + 0.05 \times 164 = 88.20[/math]

 

 

It would cost $88.20

Edited January 12, 2012 by Daedalus

[csmyth3025]

As stated, the problem is unsolvable. You apparently have the wrong wording. The delivery cost should be $0.05 per gallon per mile.

I'm embarrassed to say that you're right. I not only misread the question when I originally worked it, I also misread it again when I posted my question. The entire question - word for word from the textbook is as follows:

 

"Two gasoline distributors, A and B, are 228 miles apart on Interstate 80. A charges $0.85 per gallon and B charges $0.80 per gallon. Each charges 0.05 cents (symbol used, rather than the word) per gallon per mile for delivery. Where on Interstate 80 is the cost to the customer the same?"

(bold added by me)

 

I apologize for causing the confusion that my sloppy reading habits have spawned. I can only hope that I learn from this how important it is to read these word problems carefully!

 

Chris

[Tres Juicy]

I'm embarrassed to say that you're right. I not only misread the question when I originally worked it, I also misread it again when I posted my question. The entire question - word for word from the textbook is as follows:

 

"Two gasoline distributors, A and B, are 228 miles apart on Interstate 80. A charges $0.85 per gallon and B charges $0.80 per gallon. Each charges 0.05 cents (symbol used, rather than the word) per gallon per mile for delivery. Where on Interstate 80 is the cost to the customer the same?"

(bold added by me)

 

I apologize for causing the confusion that my sloppy reading habits have spawned. I can only hope that I learn from this how important it is to read these word problems carefully!

 

Chris

 

+1 For learning!!

[DrRocket]

I'm embarrassed to say that you're right. I not only misread the question when I originally worked it, I also misread it again when I posted my question. The entire question - word for word from the textbook is as follows:

 

"Two gasoline distributors, A and B, are 228 miles apart on Interstate 80. A charges $0.85 per gallon and B charges $0.80 per gallon. Each charges 0.05 cents (symbol used, rather than the word) per gallon per mile for delivery. Where on Interstate 80 is the cost to the customer the same?"

(bold added by me)

 

I apologize for causing the confusion that my sloppy reading habits have spawned. I can only hope that I learn from this how important it is to read these word problems carefully!

 

Chris

 

DH is right, but the omission is not much more severe than a simple typo and is readily filled in since it necessary for the problem to make sense. If that is the biggest mistake that you made today, then you are doing all right.

[csmyth3025]

DH is right, but the omission is not much more severe than a simple typo and is readily filled in since it necessary for the problem to make sense. If that is the biggest mistake that you made today, then you are doing all right.

Thanks Dr R and Tres for your gentle words. I'm hoping to not only learn the equations used in mathematics, but also how to think more analytically. In the last few months I've literally worked over four hundred questions in the first chapter of the textbook I'm using. As you might imagine, these questions started out simple and became more complex with each set of exercises. I think I've learned as much from this wrong answer as I have from the many right answers I gotten (maybe more).

 

Chris

[Tres Juicy]

Thanks Dr R and Tres for your gentle words. I'm hoping to not only learn the equations used in mathematics, but also how to think more analytically. In the last few months I've literally worked over four hundred questions in the first chapter of the textbook I'm using. As you might imagine, these questions started out simple and became more complex with each set of exercises. I think I've learned as much from this wrong answer as I have from the many right answers I gotten (maybe more).

 

Chris

 

Someone once said "victory is empty, there are lessons in defeat"