A document is to contain 80cm squared of print. The margins at the top & bottom are each 2cm wide and margins on each side are 3cm wide. What should be the dimensions (in cm) of print so that a minimum amount of paper is used?
To determine the dimensions of the print that will minimize the amount of paper used, we need to subtract the margins from the total dimensions of the document.
Given information:
- Total print area required: 80 cm²
- Top and bottom margins: 2 cm each
- Side margins: 3 cm each
Let's denote the width of the print area as "x" and the height of the print area as "y".
Calculating the total dimensions of the document:
- Width of document = print width + left margin + right margin
= x + 3 cm + 3 cm
= x + 6 cm
- Height of document = print height + top margin + bottom margin
= y + 2 cm + 2 cm
= y + 4 cm
The total area of the document can be calculated as the product of the width and height:
Total area of document (including margins) = (x + 6 cm) * (y + 4 cm)
Now, we need to subtract the area of the margins to find the area of the print:
Print area = Total area of document - 2 * (top margin * print width) - 2 * (side margin * print height)
= (x + 6 cm) * (y + 4 cm) - 2 * (2 cm * x) - 2 * (3 cm * y)
= (x + 6)(y + 4) - 4x - 6y
We are given that the print area should be 80 cm², so we can set up the equation:
(x + 6)(y + 4) - 4x - 6y = 80
Now, we can simplify the equation and solve for one variable in terms of the other:
xy + 4x + 6y + 24 - 4x - 6y = 80
xy + 24 = 80
xy = 56
Now, we can express one variable in terms of the other:
y = 56/x
Substituting this into the equation for the total area of the document:
Print area = (x + 6)(56/x + 4) - 4x - 6(56/x)
Knowing that the print area should be minimized, we can take the derivative of the print area equation with respect to x and set it equal to zero:
d(Print area)/dx = 0
After finding the critical points, we can analyze whether they correspond to a minimum or maximum.
Solving this equation analytically is somewhat complex. However, you can use numerical methods or graphing calculators to find a value for x that minimizes the print area.