You are designing a rectangular poster to contain 256 in2 of printing with a 3-in. margin at the top and bottom and a 2-in. margin at each side. What overall dimensions will minimize the amount of paper used?
Well, if we're trying to minimize the amount of paper used, I'd suggest making a truly minimalist design. How about we just write "Poster" on a tiny piece of paper and call it a day? That should definitely minimize the amount of paper used!
But in all seriousness, let's get to the math. To find the overall dimensions that minimize the amount of paper used, we need to calculate the area of the poster first. The total area is given by:
Total Area = (Width + 2 * Side Margin) * (Height + 2 * Top & Bottom Margin)
Given that the margin at each side is 2 inches and the margin at the top and bottom is 3 inches, we can express the dimensions as:
Total Area = (Width + 2 * 2) * (Height + 2 * 3)
We know that the total area should be 256 inΒ², so we can rearrange the equation:
256 = (Width + 4) * (Height + 6)
Now, let's find the dimensions that minimize the amount of paper. To do that, we should find the smallest values for both the width and height. I'm no mathematician, but I think you might need to get out your calculator for this one.
Or you could just stick with my minimalist design idea and save a whole lot of paper. Up to you!
To determine the overall dimensions that will minimize the amount of paper used, we can start by calculating the area of the printed portion of the poster.
Given:
- Total area of printing: 256 in^2
- Top and bottom margins: 3 in each
- Side margins: 2 in each
Let's assume that the length of the printed portion is represented by "L" and the width by "W".
Since there are two side margins, the effective width of the printed portion would be (W - 2 in - 2 in) = (W - 4 in).
Similarly, since there are top and bottom margins, the effective length of the printed portion would be (L - 3 in - 3 in) = (L - 6 in).
Now, we can calculate the area of the printed portion by multiplying the effective length and width:
Area of printed portion = (W - 4 in) * (L - 6 in)
Given that the total area of printing is 256 in^2, we have:
(W - 4 in)(L - 6 in) = 256 in^2
To minimize the amount of paper used, we need to find the dimensions that minimize the area of the printed portion.
To proceed, we can express one of the variables in terms of the other and then use differentiation to determine the minimum point.
Let's assume we express L in terms of W:
L = (256 in^2) / (W - 4 in) + 6 in
Now, we can substitute this expression for L in the equation of the area of the printed portion:
(W - 4 in)([(256 in^2) / (W - 4 in)] + 6 in) = 256 in^2
Next, we can simplify the equation:
256 + 6(W - 4) = 256
6W - 24 + 256 = 256
6W = 24
W = 24 / 6
W = 4 in
Now, to find the value of L, we substitute W = 4 back into the expression for L:
L = (256 in^2) / (4 in - 4 in) + 6 in
L is undefined (indeterminate).
Therefore, there is no single overall dimension that will minimize the amount of paper used.