Find the dimensions of the poster that use the least amount of board

[priscilla98]

Homework Statement

A rectangular poster is to be made out of cardboard. The margins on the top and the bottom will be 2 inches. The margins on the sides will be 1 inch. the are of the poster devoted to the printed material must be 300 square inches.

(a) To the nearest .001 inches, find the dimensions of the poster that use the least amount of cardboard.

(b) To the nearest .001 inches, find the are of the poster that uses the least amount of cardboard.

Homework Equations

Area of rectangle = length x width

The Attempt at a Solution

(a) In this problem we want to find the least amount of cardboard and we know that we will use 300 sq. in. of margins. So, the area will be the function we are trying to optimize and the amount of margins is the constraint. The two equations for these are,

Maximize: A = xy
Contraint: 300 = 2x + y

Am I right so far?

Any help will be greatly appreciated, thanks a lot

 

[HallsofIvy]

You have a basic misunderstanding. You say "we know that we will use 300 sq. in. of margins". That is not true. The problem say that "the are of the poster devoted to the printed material must be 300 square inches." That is the area that is NOT margin.

If the height and width of the poster are x inches and y inches, respectively, how would you calculate the area devoted to the printed material?

 

[priscilla98]

Okay, that's true. Wouldn't you change the length and width of the poster from inches to square inches? I know the length of this poster is 2 inches and the width is 1 inch. Therefore, wouldn't the length be 20 inches and the width is 10 inches

 

[HallsofIvy]

I'm sorry but that makes no sense at all. I don't know what you mean by "change the length and width of the poster from inches to square inches". length and width are measured in inches, not square inches. area is measured in square inches. What formula do you know for the area of a rectangle?

No, you do not "know the length of this poster is 2 inches and the width is 1 inch". Those are the widths of the margins, not the dimensions of the poster. And how did you get "20" and "10"?

Perhaps drawing a picture of the poster would help to organize your thoughts.

 

[priscilla98]

Okay, then we don't know the length of this poster yet. The area of the poster is 300 sq. in. I know the formula is A = l x w. Knowing that we know the area of the poster, can't we use this to find the dimensions of the poster. The width of the margins is 2 by 1 inches, right?

 

[HallsofIvy]

Okay, then we don't know the length of this poster yet. The area of the poster is 300 sq. in. I know the formula is A = l x w. Knowing that we know the area of the poster, can't we use this to find the dimensions of the poster. The width of the margins is 2 by 1 inches, right?

Of course, we don't know the length of the poster. The problem said to "find the dimensions of the poster"

No, the width of the margins is NOT "2 by 1 inches". There are two margins that are two inches wide and two margins that are 1 inch wide.


If the width of the poster is y inches and 1 inch on either side is taken up for margin, how much is left for the width of the printed area?

If the height of the poster is x inches and 2 inches on to and bottom are taken up for marging, how much is left for the height of the printed area?

So what would a formula for the area, in square inches, of the printed area be?