Question

william is building a garden in his backyard that is broken into two separate parts, as shown in the figure. in the square-shaped part of the garden, william is going to plant (400\text{ ft}^2) of vegetation. in the rectangular-shaped part of the garden, william is going to plant flowers.
if william wants to install a fence around the entire garden area, how much fencing will he have to install?
hint: use the formula sheet to determine the formula(s) needed to solve the problem.

Explanation:

⚡ Using what you learned: area formulas for rectangles and squares · calculating perimeter of polygons

Step 1: Find the side length of the square-shaped garden

The square-shaped part of the garden has an area of \(400\text{ ft}^2\).

Since the area of a square is given by \(A = s^2\):

$$s = \sqrt{400} = 20\text{ ft}$$

Thus, each side of the square-shaped garden is \(20\text{ ft}\).

Step 2: Determine the dimensions of the rectangular-shaped garden

From the diagram:

• The top horizontal side of the rectangular garden is labeled as \(20\text{ ft}\).
• The vertical boundary shared between the two gardens is the side of the square, which is \(20\text{ ft}\).
• The bottom portion of the rectangular garden extends downwards past the square by \(10\text{ ft}\).

Therefore, the total height (vertical side) of the rectangular garden is:

$$20\text{ ft} + 10\text{ ft} = 30\text{ ft}$$

The width (horizontal side) of the rectangular garden is \(20\text{ ft}\).

Step 3: Calculate the perimeter of the entire combined garden

To find the total fencing needed, we calculate the outer perimeter of the combined shape:

• Top outer boundary: The top of the rectangular garden (\(20\text{ ft}\)) plus the top of the square garden (\(20\text{ ft}\)) = \(40\text{ ft}\).
• Right outer boundary: The right vertical side of the square garden = \(20\text{ ft}\).
• Bottom-middle horizontal boundary: The bottom side of the square garden = \(20\text{ ft}\).
• Lower vertical boundary: The extra vertical extension of the rectangular garden = \(10\text{ ft}\).
• Bottom outer boundary: The bottom of the rectangular garden = \(20\text{ ft}\).
• Left outer boundary: The entire left vertical side of the rectangular garden = \(30\text{ ft}\).

Summing these outer sides:

$$\text{Total Fencing} = 40 + 20 + 20 + 10 + 20 + 30 = 140\text{ ft}$$

Answer:

140