- Calculate the original area of the patio: \(15 \times 12 = 180 \text{ ft}^2\).
- Define the new dimensions after extending by \(x\): \((15 + x)\) and \((12 + x)\).
- Set the new area equal to the original area plus the extension: \((15 + x)(12 + x) = 180 + 160\).
- The correct model is \(\boxed{(15 + x)(12 + x) = (15)(12) + 160}\).
Explanation
- Analyzing the Patio Expansion
The current rectangular patio has dimensions \(15 \text{ ft}\) by \(12 \text{ ft}\). The original area is calculated by multiplying these dimensions:
\[ \text{Original Area} = 15 \text{ ft} \times 12 \text{ ft} = 180 \text{ ft}^2. \]
The Davidson family intends to extend both the length and the width by the same amount, \(x \text{ ft}\). Therefore, the new length becomes \((15 + x) \text{ ft}\) and the new width becomes \((12 + x) \text{ ft}\). The new area of the expanded patio is the product of these new dimensions:
\[ \text{New Area} = (15 + x)(12 + x). \]
We are told that this expansion increases the total area by \(160 \text{ ft}^2\). Thus, the new area can also be expressed as the sum of the original area and the area increase:
\[ \text{New Area} = \text{Original Area} + 160 = (15 \times 12) + 160. \]
By setting these two expressions for the new area equal to each other, we obtain the quadratic equation that models the situation:
\[ (15 + x)(12 + x) = (15)(12) + 160. \]
- Selecting the Correct Model
We have determined that the model representing the expansion is \((15 + x)(12 + x) = (15)(12) + 160\). Looking at the provided options:
A\(A = lw\)
B\((15 + x)(12 + x) = (15)(12) + 160\)
C\((15)(12) + (x)(x) = (15)(12) + 160\)
D\(2(15 + x) + 2(12 + x) = (15)(12) + 160\)
E\((15 + x)(12 + x) = (15)(12) + 160\)
Both option B and option E correctly express the relationship between the new area and the increase in the patio's dimensions. Since these options are identical, either one accurately models the problem. Therefore, the best model for the situation is \((15 + x)(12 + x) = (15)(12) + 160\), which corresponds to option B (or E).
Examples
In real life, calculating areas for expansion is essential for home improvement projects, such as building a deck or enlarging a patio. If you have an existing space and need to determine how much you can expand it while staying within a budget or material limit, you can model the dimensions algebraically to find the unknown length or width needed to achieve your desired target area. This approach helps in planning construction and ensuring material quantities are accurate.