BR is building an addition to the west wing of the school. The addition will surround 3 sides of the building, equal width on each side The dimensions of the west wing are 20m* 72m The area of the addition will the sama as the area of the current west wing. What is the width of the new addition?

Let's denote the width of the new addition by \( x \).

The west wing has dimensions of 20 meters by 72 meters, giving it an area of:
\[
\text{Area of west wing} = 20 \times 72 = 1440 \text{ square meters}
\]

The addition will surround three sides of the west wing: two sides of length 20 meters and one side of length 72 meters. Thus, the dimensions of the new addition are \( 20 + 2x \) meters by \( 72 + 2x \) meters, excluding the original 20 by 72 meters west wing. We need to determine \( x \) such that the area of the addition equals the area of the current west wing, i.e., 1440 square meters.

The dimensions of the addition include:
- Two side extensions of width \( x \) and length 20 meters.
- One side extension of width \( x \) and length 72 meters.

Therefore, the area of the addition can be represented as the area of the larger rectangle minus the area of the west wing:
\[
(20 + 2x) \times (72 + x) - 1440 = 1440
\]

Expanding and simplifying:
\[
(20 + 2x)(72 + x) - 1440 = 1440
\]
\[
1440 + 20x + 144x + 2x^2 - 1440 = 1440
\]
\[
2x^2 + 164x = 1440
\]

Solving the quadratic equation:
\[
2x^2 + 164x - 1440 = 0
\]

Dividing by 2 to simplify:
\[
x^2 + 82x - 720 = 0
\]

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 82 \), and \( c = -720 \):
\[
x = \frac{-82 \pm \sqrt{82^2 - 4 \times 1 \times (-720)}}{2 \times 1}
\]
\[
x = \frac{-82 \pm \sqrt{6724 + 2880}}{2}
\]
\[
x = \frac{-82 \pm \sqrt{9604}}{2}
\]
\[
x = \frac{-82 \pm 98}{2}
\]

This gives two solutions:
\[
x = \frac{16}{2} = 8 \quad \text{and} \quad x = \frac{-180}{2} = -90
\]

Since a width cannot be negative, we have:
\[
x = 8
\]

Thus, the width of the new addition is 8 meters.

Would you like more details or have any questions?

Here are 5 more questions you might want to explore:
1. How do you solve quadratic equations using the quadratic formula?
2. What is the method of completing the square in solving quadratics?
3. How do you factorize a quadratic equation?
4. What are some applications of quadratic equations in real-life scenarios?
5. How do you find the area of composite figures in geometry?

**Tip:** Always check the context of a problem to ensure the solutions make sense (e.g., non-negative dimensions in geometry problems).

Learn how to calculate the width of a school addition surrounding the west wing using geometric principles and quadratic equations. This problem involves finding the dimensions of an added section equal to the area of the existing west wing. Suitable for students in Grades 7-9.

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