A rectangular poster is to have 50 squared inches for printed matter. It is to have a 1 inch. margin on each side and a 2 inch. margin at the top and bottom. Find the dimensions of the poster so that the amount of paper used is minimized.
Horizonatal width=
Vertical Height=
A rectangular poster is to have 50 squared inches for printed matter. It is to have a 1 inch. margin on each side and a 2 inch. margin at the top and bottom. Find the dimensions of the poster so that the amount of paper used is minimized.
Horizonatal width=
Vertical Height=
Let the printed area xy=50
The total area is
a = (x+2)(y+4) = (x+2)(50/x + 4)
= 4x + 58 + 100/x
so, find where da/dx = 0 (and x>0)
To find the dimensions of the poster that minimize the amount of paper used, we need to optimize the area of the poster.
Let's start by defining the dimensions of the poster. Since there is a 1 inch margin on each side, the actual printed area will be reduced by 2 inches in total from the width. Similarly, since there is a 2 inch margin at the top and bottom, the actual printed area will be reduced by 4 inches from the height.
Let's call the width of the printed area as x (in inches) and the height of the printed area as y (in inches).
Given the dimensions of the margin, the total width of the poster would be x + 2 (margin on the left) + 2 (margin on the right), and the total height would be y + 4 (margin at the top) + 4 (margin at the bottom).
Now, we need to find the area of the printed area by subtracting the total area reduced by the margins from the total area of the poster.
Area of the printed area = (x)(y)
Total area of the poster = (x + 4)(y + 8)
The problem states that the area of the printed matter is to be 50 square inches. So, we have the equation:
(x)(y) = 50
We want to minimize the amount of paper used, which means we want to minimize the total area of the poster.
To find the dimensions of the poster that minimize the amount of paper used, we need to solve the equation for the area of the printed area and substitute it into the equation for the total area of the poster:
(x)(y) = 50
(x + 4)(y + 8) = Total area of the poster
Now, you can solve these two equations simultaneously to find the values of x and y that minimize the total area of the poster.