Optimization

Date: 12/13/95 at 11:10:44
From: Anonymous
Subject: Optimization in Calculus

Dear teachers,

I am a AP Caluculus student at a military school in Augsburg, Germany.
On Wednesday, December 13, 1995, I took a chapter test in my class.
Having had a problem on 2 of the questions, I was wondering if you
could explain how to complete the problems.

1. Find the equation for the graph that passes through the pt. (-1,-4) 
with slope 2 given

 d^2y/dx^2 = 3x/4

I am totally lost on this one.

2. You are designing a poster to contain 1200 sq. in of printing with 
margins of 4 in. each at the top and bottom and 3 in. at each side.  
What overall dimensions will minimize the amount of paper used.

On this one, I've figured to start it out by having:

(x-6)(x-8)=1200

Help me out from here.

Your answer will be greatly appreciated.

Ara Donabedian

Date: 12/15/95 at 11:24:12
From: Doctor Ken
Subject: Re: Optimization in Calculus

Hello!

>1. Find the equation for the graph that passes through the pt. (-1,-4) 
>with slope 2 given
>
> d^2y/dx^2 = 3x/4
>
>I am totally lost on this one.

Okay, well let's think about what that statement means.  Essentially, 
the d^2y/dx^2 = 3x/4 part means that the second derivative of our 
function is 3x/4.  So what we're going to have to do to find the answer 
is take two antiderivatives of 3x/4.  Well, if you take one 
antiderivative you get 3x^2/8 + c, and if you take another you get x^3/8 
+ cx + d.  Now you'll have to figure out how to use the 
information about slope and point that it goes through (hint: slope = 
derivative) to find c and d.

>2. You are designing a poster to contain 1200 sq. in of printing with 
>margins of 4 in. each at the top and bottom and 3 in. at each side.  
>What overall dimensions will minimize the amount of paper used.
>
>On this one, I've figured to start it out by having:
>
>(x-6)(x-8)=1200

Actually, what you're going to want is (x-6)(y-8) = 1200, letting x and y
be the length and width of the paper (it's not necessarily a square).  
Then the function that you want to minimize is the area of the paper, 
Area = xy.  The way the first equation is going to fit into the picture 
is that you're going to solve for x in terms of y (or y in terms of x, 
it doesn't really matter) and then plug that into the Area equation, 
then minimize.  Good luck!

-Doctor Ken,  The Math Forum