Question
A rectangular stained glass window is 2 feet by 4 feet. A clear glass border is constructed around the stained glass window. The width of the border is equal and was made out of 7 square feet of clear glass. What is the width of the border?
Answers
GPT 3.5
To find the width of the border, we first need to calculate the total area of the entire window including the border.
Area of the stained glass window = 2 ft x 4 ft = 8 square feet
Let x be the width of the border
Total area of the window with the border = (2+2x)(4+2x)
Total area = (2+2x)(4+2x) = 8 + 16x + 4x + 8x^2
Total area = 8 + 20x + 8x^2
Now we know that the border is made out of 7 square feet of clear glass, so:
Total area - Area of stained glass window = Area of clear glass border
8 + 20x + 8x^2 - 8 = 7
20x + 8x^2 = 7
8x^2 + 20x - 7 = 0
Solving for x using the quadratic formula:
x = (-b ± √(b^2-4ac)) / 2a
x = (-20 ± √(20^2-4*8*(-7))) / 2*8
x = (-20 ± √(400+224)) / 16
x = (-20 ± √624) / 16
x = (-20 ± 24.98) / 16
x ≈ (-20 + 24.98) / 16 ≈ 0.31 feet
x ≈ (-20 - 24.98) / 16 ≈ -2.80 feet
Since width cannot be negative, the width of the border is approximately 0.31 feet.
Area of the stained glass window = 2 ft x 4 ft = 8 square feet
Let x be the width of the border
Total area of the window with the border = (2+2x)(4+2x)
Total area = (2+2x)(4+2x) = 8 + 16x + 4x + 8x^2
Total area = 8 + 20x + 8x^2
Now we know that the border is made out of 7 square feet of clear glass, so:
Total area - Area of stained glass window = Area of clear glass border
8 + 20x + 8x^2 - 8 = 7
20x + 8x^2 = 7
8x^2 + 20x - 7 = 0
Solving for x using the quadratic formula:
x = (-b ± √(b^2-4ac)) / 2a
x = (-20 ± √(20^2-4*8*(-7))) / 2*8
x = (-20 ± √(400+224)) / 16
x = (-20 ± √624) / 16
x = (-20 ± 24.98) / 16
x ≈ (-20 + 24.98) / 16 ≈ 0.31 feet
x ≈ (-20 - 24.98) / 16 ≈ -2.80 feet
Since width cannot be negative, the width of the border is approximately 0.31 feet.