A graphic designer is asked to create a movie poster with a 162 in2 photo surrounded by a 4 in border at the top and bottom and a 2 in border on each side. What overall dimensions for the poster should the designer choose to use the least amount of paper?
13/8 by 576/13
To determine the overall dimensions for the movie poster with the least amount of paper, we can add the width and height of the photo to the respective borders.
Let's assume the width of the photo is x inches. The width of the poster would then be x + 2 inches on each side.
Similarly, let's assume the height of the photo is y inches. The height of the poster would be y + 4 inches at the top and bottom.
We can then calculate the area of the poster using the formula:
Area = Width x Height
Area = (x + 2 + 2) inches * (y + 4 + 4) inches
Area = (x + 4) inches * (y + 8) inches
Since the area of the poster should be minimized, we need to find the dimensions that minimize the value of (x + 4) inches * (y + 8) inches.
To find the minimum area, we take the derivative of the area with respect to either x or y and set it equal to zero. However, since we know the poster must accommodate a 162 in² photo, we can set up the following equation:
162 = x * y
Solving for y, we get:
y = 162 / x
Substituting this value of y into the equation for the area, we have:
Area = (x + 4) inches * (162 / x + 8) inches
Next, we multiply these binomials:
Area = 162 + 8x + 4(162 / x) + 4x
Expanding further:
Area = 162 + 4x + (648 / x) + 4x
Combining like terms, we have:
Area = 8x + (648 / x) + 162
To find the minimum area, we take the derivative of the area with respect to x and set it equal to zero:
d(Area) / dx = 8 - (648 / x²) = 0
To simplify, multiply both sides by x²:
8x² - 648 = 0
Now, solve for x:
8x² = 648
x² = 648 / 8
x² = 81
x = √(81)
x = 9
Now that we have the width of the photo, we can calculate the height using:
y = 162 / x
y = 162 / 9
y = 18
Therefore, the optimal dimensions for the movie poster are 9 inches by 18 inches, resulting in the least amount of paper usage.
To determine the overall dimensions for the movie poster that would use the least amount of paper, we need to calculate the area of the poster with the given photo and borders.
Let's start by calculating the total area of the photo and borders. We know that the photo has an area of 162 square inches. To find the total area of the poster, we need to add the areas of the photo and each border.
The border at the top and bottom has a height of 4 inches, so let's add twice that to the height of the photo: 4 in + 4 in = 8 in.
The border on each side has a width of 2 inches, so let's add twice that to the width of the photo: 2 in + 2 in = 4 in.
Now we have the dimensions of the photo with the borders: 4 in (width) and 8 in (height).
To calculate the area of the photo with the borders, we multiply the width by the height: 4 in * 8 in = 32 in².
Finally, we add this area to the area of the photo to find the total area of the poster: 32 in² + 162 in² = 194 in².
Therefore, the graphic designer should choose overall dimensions for the movie poster that have a total area of 194 square inches to use the least amount of paper.
If the photo's dimensions are
width = x
height = y
xy = 162, so y = 162/x
Then we have total poster area is
a = (x+4)(y+8) = (x+4)(162/x + 8)
find x such that da/dx = 0