Let's find the dimensions of a storm window that has an area of 448 square inches, knowing that the window is 12 inches higher than it is wide.
1. Define Variables:
Let [tex]\( w \)[/tex] be the width of the window in inches.
Let [tex]\( h \)[/tex] be the height of the window in inches.
2. Relate Height and Width:
Given that the height is 12 inches more than the width, we have:
[tex]\[
h = w + 12
\][/tex]
3. Formula for Area:
The area of the window is given by the product of its width and height:
[tex]\[
\text{Area} = w \times h
\][/tex]
Given the area is 448 square inches, we substitute [tex]\( h \)[/tex] from step 2 into the area formula:
[tex]\[
w \times (w + 12) = 448
\][/tex]
4. Setting up the Quadratic Equation:
Expand and simplify the equation:
[tex]\[
w^2 + 12w = 448
\][/tex]
Rewrite it in standard quadratic form:
[tex]\[
w^2 + 12w - 448 = 0
\][/tex]
5. Solve the Quadratic Equation:
To solve the quadratic equation [tex]\( w^2 + 12w - 448 = 0 \)[/tex], we find the roots of the equation. The solutions for [tex]\( w \)[/tex] are:
[tex]\[
w = 16 \quad \text{and} \quad w = -28
\][/tex]
We discard the negative value since width cannot be negative. So, we have:
[tex]\[
w = 16
\][/tex]
6. Find the Height:
Now, calculate the height using the width:
[tex]\[
h = w + 12
\][/tex]
Substituting [tex]\( w = 16 \)[/tex]:
[tex]\[
h = 16 + 12 = 28
\][/tex]
7. Conclusion:
The dimensions of the window are:
- Width: 16 inches
- Height: 28 inches
So, the window has a width of 16 inches and a height of 28 inches, making the area 448 square inches.
