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# Problem Solving - Math?

The number of miles M that a certain automobile can travel on one gallon of gas at a speed of v mph is given by the following formula:

M=-(1/30)v^(2)+(5/2)v for 0 < v < 70

a. For the consumption of one gallon of gasoline, find the most economical speed.

b. Find the maximum number of miles that can be driven at the most economical speed.

I've always been bad at beginning these type of problems. I don't really need an answer unless it isn't too much trouble because I'd have something to check my work with. I'm just wondering how to go about beginning the processes of both of these problems.

### 2 Answers

- TonyLv 68 years agoFavorite Answer
You have the function:

- v^2/30 + 5v/2 0 < v < 70

You should recognize this an an "inverted" parabola,probably given a textbook problem, with the vertex somewhere in the range 0 < x < 70

I presume this is a maxima/minima problem using differential calculus. Let's differentiate to find the vertex, although you COULD go back to high school algebra and determine the vertex by algebraic methods.

-2v/30 + 5/2 = f(x)

-v/15 + 5/2 = f(x)

now, set the function equal to zero and solve for x:

v/15 = 5/2

v = 75/2 = 37.5 mph

NOW, for part B, go back to the original function

M = -v^2/30 + 5v/2 and plug in 37.5 for "v"

always,

tony

- Anonymous8 years ago
Hello Swag,

I have posted a full solution to your problem at one of the math help forums I help to moderate so that I may give you an easy to read explanation using LaTeX:

http://www.mathhelpboards.com/f2/swags-question-ya...

edit: I made the assumption based on your remarks that you were using pre-calculus methods.