A poster is to have an area of 160 in2 with 1 inch margins at the bottom and sides and a 7 inch margin at the top. What dimensions will give the largest printed area? (Give your answers correct to one decimal place.)
If the dimensions are x wide and y high, then
y = 160/x
The printed area is
a = (x-2)(y-1-7) = (x-2)(160/x - 8) = 176-8x-320/x
da/dx = -8+320/x^2
da/dx=0 when x=2√10
Now figure y and you're done.
To find the dimensions that will give the largest printed area, we need to maximize the area of the poster. Let's first find the dimensions of the printable area.
The poster has 1-inch margins at the bottom and sides, and a 7-inch margin at the top. So, the printable height (height excluding margins) would be the total height minus the combined margin height:
Printable Height = Total Height - Top Margin = (Total Height) - 7
Similarly, the printable width (width excluding margins) would be the total width minus the combined margin width:
Printable Width = Total Width - (2 * Side Margin) = (Total Width) - (2 * 1)
Now, we can express the area of the poster in terms of the printable dimensions:
Area = Printable Height * Printable Width
We are given that the area of the poster is 160 in^2. Substituting the expressions for height and width, we have:
160 = (Total Height - 7) * (Total Width - 2)
To find the dimensions that give the largest printed area, we need to maximize this area function. One way to do this is by finding the critical points of the area equation. We can do this by finding the derivative of the area function with respect to one of the variables (either height or width) and setting it equal to zero:
d(Area) / d(Height) = (Total Width - 2) * d(Total Height - 7) / d(Total Height) + (Total Height - 7) * d(Total Width - 2) / d(Total Height) = 0
Simplifying this equation will give us the relationship between Total Height and Total Width that maximizes the area. Finally, we can substitute this into the area equation to determine the dimensions that give the largest printed area.