Answer
$9$ ft $\times$ $4.5$ ft
Work Step by Step
We are given the following about the door's dimensions:
Length = 7 ft 6 inches
Width = 3 ft
First, we should convert 7 ft 6 inches to feet completely to make it easier.
7 ft 6 inches = 7.5 ft
Then, we are given that the same amount is added to each dimension, increasing the area by $18ft^2$. The amount added to each dimension can be represented as $x$.
The original area of the door can be given by the formula, Length $\times$ Width.
Area = Length $\times$ Width
Area = 7.5 x 3
Area = $22.5 ft^2$
The area was increased by $18ft^2$ after $x$ amount was added to the dimensions, so the new area is;
22.5 + 18 = $40.5ft^2$
The new dimensions are:
Length = $7.5 + x$ ft
Width = $3+x$ ft
We can substitute these values into the formula for area of a rectangle to form a quadratic equation!
Area = Length $\times$ Width
$40.5 = (7.5 + x)(3+x)$
$40.5 = x^2+3x+7.5x+22.5$
$40.5 = x^2+10.5x+22.5$
(We can multiply both sides of the equation by 2 to get rid of the decimals)
$40.5 \times 2 = (x^2+10.5x+22.5)\times 2$
$81=2x^2+21x+45$ (Then, Subtract 81 from both sides)
$2x^2+21x-36=0$
We can use the quadratic formula to solve this equation!
$x = \frac{-b±\sqrt {b^2-4ac}}{2a}$
$x_{1} = \frac{-21+\sqrt {21^2-4(2)(-36)}}{2(2)}$, $x_{2} = \frac{-21-\sqrt {21^2-4(2)(-36)}}{2(2)}$
$x_{1} = \frac{-21+27}{4}$, $x_{2} = \frac{-21-27}{4}$
$x_{1} = 1.5$, $x_{2} = -12$
$x$ cannot be negative in this case as we are finding the dimensions of a door, so $x = 1.5$
New Length = $7.5 + (1.5) $
= $9$ ft
New Width = $3 + (1.5)$
= $4.5$ ft
