Question

problem-solving

5 a rectangular garden bed is 12 m long and 5 m wide.
a find the area of the garden bed.
b the length is increased by \\(x\\) m and the width is decreased by \\(y\\) m. find the new length and width of the garden.
c write an expression for the area of the new garden bed.

Explanation:

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<reasoning>

Calculate the initial area

The garden bed is a rectangle.
The area is length times width.
\[
\text{Area} = 12 \times 5 = 60\text{ m}^2
\]

Determine the new dimensions

The original length is \(12\text{ m}\) and is increased by \(x\text{ m}\).
The original width is \(5\text{ m}\) and is decreased by \(y\text{ m}\).
\[

$$\begin{aligned} \text{New Length} &= (12 + x)\text{ m} \\ \text{New Width} &= (5 - y)\text{ m} \end{aligned}$$

\]

Formulate the new area expression

The new area is the product of the new dimensions.
\[
\text{New Area} = (12 + x)(5 - y)\text{ m}^2
\]
</reasoning>

<answer>
a The area of the garden bed is \(60\text{ m}^2\).

b The new length is \((12 + x)\text{ m}\) and the new width is \((5 - y)\text{ m}\).

c The expression for the area of the new garden bed is \((12 + x)(5 - y)\text{ m}^2\).
</answer>

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Answer:

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<reasoning>

Calculate the initial area

The garden bed is a rectangle.
The area is length times width.
\[
\text{Area} = 12 \times 5 = 60\text{ m}^2
\]

Determine the new dimensions

The original length is \(12\text{ m}\) and is increased by \(x\text{ m}\).
The original width is \(5\text{ m}\) and is decreased by \(y\text{ m}\).
\[

$$\begin{aligned} \text{New Length} &= (12 + x)\text{ m} \\ \text{New Width} &= (5 - y)\text{ m} \end{aligned}$$

\]

Formulate the new area expression

The new area is the product of the new dimensions.
\[
\text{New Area} = (12 + x)(5 - y)\text{ m}^2
\]
</reasoning>

<answer>
a The area of the garden bed is \(60\text{ m}^2\).

b The new length is \((12 + x)\text{ m}\) and the new width is \((5 - y)\text{ m}\).

c The expression for the area of the new garden bed is \((12 + x)(5 - y)\text{ m}^2\).
</answer>

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