A photo is printed on a 20-inch by 24-inch piece of paper. The photo covers 320 square inches and has a uniform border. What is the width of the border?
if the width is w, then
(20-2w)(24-2w) = 320
To calculate the width of the border, we need to subtract the area of the photo from the total area of the paper and then divide the remaining area by the sum of the width and length of the photo. Here's how you can do it step by step:
1. Calculate the total area of the paper:
Total Area = Width × Length
Total Area = 20 inches × 24 inches
Total Area = 480 square inches
2. Subtract the area of the photo from the total area to find the area of the border:
Border Area = Total Area - Photo Area
Border Area = 480 square inches - 320 square inches
Border Area = 160 square inches
3. Let's assume the width of the border is x inches. Since the width of the photo is 20 inches and the length is 24 inches, the overall width of the paper (including the border) would be (20 + 2x) inches and the overall length would be (24 + 2x) inches.
4. To find the area of the border, we can multiply the width (20 + 2x) by the length (24 + 2x) and subtract the area of the photo (320 square inches):
Border Area = (20 + 2x) × (24 + 2x) - 320 square inches
160 square inches = (480 + 88x + 4x^2) - 320 square inches
160 square inches = 160 + 88x + 4x^2 - 320 square inches
0 = 88x + 4x^2
5. Rearrange the equation to form a quadratic equation:
4x^2 + 88x = 0
6. Factor out common terms:
4x(x + 22) = 0
7. Set each part equal to zero and solve for x:
4x = 0 or x + 22 = 0
If 4x = 0, then x = 0.
If x + 22 = 0, then x = -22.
8. Since the width of the border cannot be negative, we discard x = -22.
Therefore, the width of the border is 0 inches.