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.)

A1 = 3 * 7.5 = 22.5 Ft^2.

A2 = 22.5 + 5.5 = 28 Ft^2.
(3+x) * (7.5+x) = 28,
22.5 + 3x + 7.5x + x^2 = 28,
x^2 + 10.5x - 5.5 = 0.
Use Quadratic formula:
X = 0.5 Ft.
Width = 3 + 0.5 Ft. = 3.5 Ft.
Ht. = 7.5 + 0.5 = 8 Ft.

To solve this problem, we need to set up equations based on the given information.

Let's start by finding the area of the original door. The formula for calculating the area of a rectangle is length multiplied by width. In this case, the length is 3 ft and the width is 7 ft 6 in.

To convert the width from feet and inches to feet, we need to remember that 1 ft is equal to 12 in. Therefore, 7 ft 6 in can also be written as 7 ft + (6/12) ft.

Now let's calculate the area of the original door:

Area = Length × Width
Area = 3 ft × (7 ft + (6/12) ft)
Area = 3 ft × (7 ft + 0.5 ft)
Area = 3 ft × 7.5 ft
Area = 22.5 ft²

Next, let's find the dimensions of the new door by adding the same amount to each dimension.

Let's assume we add 'x' feet to both the width and height of the door.

According to the given information, the new area is increased by 5.5 ft². Therefore, the new area can be calculated as:

New Area = Area + 5.5 ft²

Substituting the values we know:

New Area = 22.5 ft² + 5.5 ft²
New Area = 28 ft²

Now, let's set up the equation based on the formula for the area of a rectangle.

New Area = New Length × New Width

Since we are adding the same value to both the length and width, we can write the equations as:

(Length + x) × (Width + x) = 28 ft²

Substituting the known values:

(3 ft + x) × (7.5 ft + x) = 28 ft²

Now, we solve this quadratic equation to find the dimension of the new door:

(3 ft + x) × (7.5 ft + x) = 28 ft²
22.5 ft² + 3x + 7.5x + x² = 28 ft²
x² + 10.5x + 22.5 ft² = 28 ft²
x² + 10.5x - 5.5 ft² = 0

To solve this quadratic equation, we can either factor it or use the quadratic formula. Factoring it might not be straightforward, so let's use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In our equation, a = 1, b = 10.5, and c = -5.5.

x = (-(10.5) ± √((10.5)² - 4(1)(-5.5))) / (2(1))

Simplifying further:

x = (-10.5 ± √(110.25 + 22)) / 2
x = (-10.5 ± √(132.25)) / 2
x = (-10.5 ± 11.5) / 2

This equation has two solutions:

x₁ = (-10.5 + 11.5) / 2 = 0.5
x₂ = (-10.5 - 11.5) / 2 = -11

Since the dimensions of a door cannot be negative, we discard the solution x₂ = -11.

Therefore, the value of x is 0.5 ft.

To find the new dimensions of the door, we add 0.5 ft (or 6 in) to both the width and height:

New Width = 3 ft + 0.5 ft = 3.5 ft
New Height = 7.5 ft + 0.5 ft = 8 ft

Therefore, the dimensions of the new door are 3.5 ft wide by 8 ft high.