A graphic designer is asked to create a movie poster with a 128 in^2 photo surrounded by a 4 in border at the top and bottoms 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?

To determine the overall dimensions for the movie poster that will use the least amount of paper, we can break down the problem into a series of steps:

Step 1: Calculate the dimensions of the photo area.
The graphic designer is asked to create a movie poster with a 128 in^2 photo. Since the photo area is rectangular, we need to find its dimensions. Let's assume the length of the photo is "x" inches. Then, the width of the photo can be calculated using the given area as follows:

Area of photo = length x width
128 in^2 = x in x width in
width in = 128 in^2 / x in

Step 2: Calculate the total width of the poster.
The total width of the poster can be determined by adding the photo width and the borders on each side:

Total width of poster = width in + 2 in (left border) + 2 in (right border)
= width in + 4 in

Step 3: Calculate the total height of the poster.
The total height of the poster can be calculated by adding the photo height and the borders at the top and bottom:

Total height of poster = height in + 4 in (top border) + 4 in (bottom border)
= height in + 8 in

Step 4: Calculate the overall dimensions of the poster.
The overall dimensions of the poster can be calculated by combining the total width and total height:

Overall dimensions of poster = Total width x Total height
= (width in + 4 in) x (height in + 8 in)

Step 5: Minimize the area of the overall dimensions.
To minimize the amount of paper used, we need to find the dimensions that minimize the overall area. Since the overall area is given by the expression (width in + 4 in) x (height in + 8 in), we can look for the dimensions that minimize this expression.

To do this, we could differentiate the expression with respect to x, set the derivative equal to zero, and solve for x to find the optimal value. However, to simplify the problem, we can observe that the expression will be smallest when x is as small as possible (to minimize the photo area) while still maintaining the given area of 128 in^2.

Therefore, to minimize the amount of paper used, the designer should choose the overall dimensions of the poster as follows:

Width: width in + 4 in
Height: height in + 8 in

By using these dimensions, the graphic designer can create the movie poster that uses the least amount of paper while keeping the required photo area of 128 in^2.

To determine the overall dimensions for the movie poster that will use the least amount of paper, we need to consider the dimensions of the photo and the borders.

Given:
Area of the photo = 128 in^2
Top and bottom border = 4 in each
Side borders = 2 in each

Let's calculate the dimensions of the photo by subtracting the border sizes from the overall dimensions:
Width of the photo = Width of the poster - 2 × side border
= Width of the poster - 2 × 2 in
= Width of the poster - 4 in

Height of the photo = Height of the poster - 2 × top/bottom border
= Height of the poster - 2 × 4 in
= Height of the poster - 8 in

Now, we know that the area of the photo is given by:
Area of the photo = Width of the photo × Height of the photo
128 in^2 = (Width of the poster - 4 in) × (Height of the poster - 8 in)

To minimize the overall dimensions, we aim to minimize the area of the poster. Since the width and height are both positive values, we can conclude that minimizing the area is equivalent to minimizing the product (Width - 4) × (Height - 8).

We can solve this problem by finding the factors of 128 in^2 and identifying the combination that minimizes the product. The factors of 128 in^2 are: 1, 2, 4, 8, 16, 32, 64, 128.

Now we will calculate the various widths and heights that satisfy the given conditions:
Width of the poster (W) = factors of 128 + 4 in
Height of the poster (H) = factors of 128 + 8 in

Let's calculate the area for each combination of dimensions:
(W1 - 4 in) × (H1 - 8 in) = Area1
(W2 - 4 in) × (H2 - 8 in) = Area2
(W3 - 4 in) × (H3 - 8 in) = Area3
...
(Wn - 4 in) × (Hn - 8 in) = Arean

Compare the areas and select the combination of dimensions with the smallest area (Arean). This will give us the overall dimensions of the movie poster that will use the least amount of paper.

let the width of the photo be x in

then the length of the photo is 128/x in

width of poster = x+8
length of poster = 128/x + 8

Area = (x+8)(128/x + 8)
= 128 + 8x + 1024/x + 64
d(area) = 8 - 1024/x^2
= 0 for a min of area

8 = 1024/x
8x^2 = 1024
x^2 =128
x = √128 = 8√2

carry on