Understanding the Rectangular Garden Problem

We are given a problem about a rectangular garden. We need to find its area.

• The half perimeter of the garden is given as 42 cm.
• The length of the garden is 8 cm more than its width.

Let's denote the length of the rectangle by $l$ and the width by $w$. The units for length and width are centimeters (cm).

Calculating Garden Dimensions

First, we need to find the actual length and width of the garden using the information provided.

1. Relating Half Perimeter to Length and Width: The perimeter of a rectangle is calculated as $2 \times (length + width)$, or $2(l+w)$. Half the perimeter is therefore $l+w$. We are given that half the perimeter is 42 cm. So, we have the equation:

   $$l + w = 42 \quad (1)$$

2. Relating Length and Width: We are told that the length is 8 cm more than the width. This can be written as:

   $$l = w + 8 \quad (2)$$

3. Solving for Width ($w$): Now we can substitute the expression for $l$ from equation (2) into equation (1):

   $$(w + 8) + w = 42$$

   Combine the terms with $w$:

   $$2w + 8 = 42$$

   Subtract 8 from both sides:

   $$2w = 42 - 8$$

   $$2w = 34$$

   Divide by 2:

   $$w = \frac{34}{2}$$

   $$w = 17 \text{ cm}$$

4. Solving for Length ($l$): Now that we have the width, we can find the length using equation (2):

   $$l = w + 8$$

   $$l = 17 + 8$$

   $$l = 25 \text{ cm}$$

So, the width of the garden is 17 cm and the length is 25 cm.

Finding the Garden Area

The area of a rectangle is found by multiplying its length by its width.

$$Area = l \times w$$

Substitute the values we found for $l$ and $w$:

$$Area = 25 \text{ cm} \times 17 \text{ cm}$$

Now, let's calculate the product:

$$Area = 425 \text{ cm}^2$$

The area of the rectangular garden is 425 square centimeters.