The dimensions of a door are 3 ft wide by 7 ft 6 in high. If the same amount is added to each dimension of the door, the area is increased by 5.5 ft2. Find the dimensions of the new door. (Round your answer to one decimal place.)
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To solve this problem, we can set up an equation based on the given information.
Let's start by finding the area of the original door. The formula for the area of a rectangle is length multiplied by width.
Given:
Original width = 3 ft
Original height = 7 ft 6 in
First, convert the height to feet:
7 ft 6 in = 7 ft + 6 in
= 7 ft + 6/12 ft
= 7 ft + 0.5 ft
= 7.5 ft
Now we can calculate the area of the original door:
Original area = Original width × Original height
= 3 ft × 7.5 ft
= 22.5 ft²
According to the problem, the same amount is added to each dimension of the door, resulting in an area increase of 5.5 ft². Let's assume this additional amount as "x".
So, the new dimensions of the door will be:
New width = Original width + x
New height = Original height + x
And the new area will be:
New area = (Original width + x) × (Original height + x)
We are given that the new area is increased by 5.5 ft² compared to the original area. So, we can set up the following equation:
New area - Original area = 5.5 ft²
[(Original width + x) × (Original height + x)] - Original area = 5.5 ft²
Substituting the given values:
[(3 ft + x) × (7.5 ft + x)] - 22.5 ft² = 5.5 ft²
To solve this equation, we can expand the expression, simplify, and solve for x.
(3 ft + x) × (7.5 ft + x) - 22.5 ft² = 5.5 ft²
(3 × 7.5) ft² + (3 × x) ft + (x × 7.5) ft + (x × x) ft² - 22.5 ft² = 5.5 ft²
22.5 ft² + 3x ft + 7.5x ft + x² ft² - 22.5 ft² = 5.5 ft²
10.5x ft + x² ft² = 5.5 ft²
x² + 10.5x - 5.5 = 0
Now we have a quadratic equation. To solve for x, we can either factor this equation or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
For our equation, a = 1, b = 10.5, and c = -5.5. Plugging in these values:
x = (-(10.5) ± √((10.5)² - 4(1)(-5.5))) / (2(1))
x = (-10.5 ± √(110.25 + 22)) / 2
x = (-10.5 ± √(132.25)) / 2
x = (-10.5 ± 11.5) / 2
Using both the positive and negative values to find two possible solutions:
x₁ = (-10.5 + 11.5) / 2 = 0.5 ft
x₂ = (-10.5 - 11.5) / 2 = -11 ft
Since the dimensions cannot be negative, we discard x₂ = -11 ft as a valid solution.
Therefore, the additional amount added to each dimension of the door is 0.5 ft.
Finally, we can calculate the new dimensions:
New width = Original width + x
= 3 ft + 0.5 ft
= 3.5 ft
New height = Original height + x
= 7.5 ft + 0.5 ft
= 8 ft
So, the dimensions of the new door are 3.5 ft wide by 8 ft high.