You are building a frame for a photograph that is 8 inches by 11 inches. The width of the frame must be uniform around the picture and you have 42 square inches of wood to use. What is the width of the frame?
photo area = 88 in^2
outside dimension = 8+2x by 11+2x
so we want
(2x+8)(2x+11) - 88 = 42
4 x^2 + 38 x - 42 = 0
2 x^2 + 19 x - 21 = 0
(2x + 21)(x-1) = 0
x = 1 inch
To find the width of the frame, we first need to calculate the area of the photograph. The area of a rectangle is calculated by multiplying its length by its width.
Given that the photograph is 8 inches by 11 inches, the area of the photograph is:
Area of the photograph = length × width
= 8 inches × 11 inches
= 88 square inches
We also know that the frame's width is uniform around the picture, meaning that the frame will have the same width on all sides. Let's denote the width of the frame as "x" inches.
Now, to determine the width of the frame, we have to subtract the area of the photograph from the total area of the frame (which includes the photograph and the frame). We are given that the total area of the frame is 42 square inches.
Total area of the frame = area of the photograph + area of the frame
= 88 square inches + area of the frame
Given that the total area of the frame is 42 square inches, we can set up the following equation:
42 square inches = 88 square inches + area of the frame
To solve for the area of the frame, we rearrange the equation:
area of the frame = 42 square inches - 88 square inches
= -46 square inches
Since the area of the frame cannot be negative, it means that the total area of the frame is less than the area of the photograph. In this case, it is not possible to create a frame with the given specifications and the given amount of wood.
To find the width of the frame, we can subtract the area of the photograph from the total area of the frame (including the photograph).
Let's label the width of the frame as "x". Since the frame is uniform around the picture, we can add 2x to each dimension of the photograph to get the dimensions of the frame:
Width of frame = 8 + 2x
Length of frame = 11 + 2x
The area of the photograph is 8 inches multiplied by 11 inches, which gives us 88 square inches.
Area of the frame + photograph = Total area of wood available
(8 + 2x) * (11 + 2x) = 42
Expanding the equation, we get:
88 + 18x + 22x + 4x^2 = 42
Combining like terms, we have:
4x^2 + 40x + 88 = 42
Subtracting 42 from both sides, we get:
4x^2 + 40x + 46 = 0
Next, we can divide all terms by 2 to simplify the equation:
2x^2 + 20x + 23 = 0
This equation cannot be factored easily, so we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 20, and c = 23. Substituting these values into the quadratic formula, we have:
x = (-20 ± √(20^2 - 4 * 2 * 23)) / (2 * 2)
x = (-20 ± √(400 - 184)) / 4
x = (-20 ± √216) / 4
x = (-20 ± 14.7) / 4
Solving for both possible values of x, we get:
x ≈ (-20 + 14.7) / 4 ≈ -5.3 / 4 ≈ -1.33
x ≈ (-20 - 14.7) / 4 ≈ -34.7 / 4 ≈ -8.68
Since we're looking for the width of the frame, we know that it must be a positive value. Therefore, the width of the frame is approximately 1.33 inches.