Question

dan is building a fence around a rectangular section of his property. the design for the fenced region is shown below.

what are the perimeter and area of the fenced region?

the perimeter of the fenced region is feet.

the area of the fenced region is square feet.

options:
(400sqrt{3}) , (80sqrt{6}) , (400sqrt{6}) , (40 + 40sqrt{6}) , (1,200) , (20 + 20sqrt{3}) , (40 + 40sqrt{3})

Explanation:

⚡ Using what you learned: Roots and Radicals

Step 1: Calculate the perimeter

The perimeter \( P \) of a rectangle is calculated using the formula:

$$ P = 2l + 2w $$

Given:

• Length \( l = 20\sqrt{3}\text{ ft} \)
• Width \( w = 20\text{ ft} \)

Substitute the values into the formula:

$$ P = 2(20\sqrt{3}) + 2(20) $$

$$ P = 40\sqrt{3} + 40 $$

Rearranging the terms to match the options:

$$ P = 40 + 40\sqrt{3}\text{ feet} $$

Step 2: Calculate the area

The area \( A \) of a rectangle is calculated using the formula:

$$ A = l \times w $$

Substitute the values into the formula:

$$ A = (20\sqrt{3}) \times 20 $$

$$ A = 400\sqrt{3}\text{ square feet} $$

Answer:

The perimeter of the fenced region is \( 40 + 40\sqrt{3} \) feet.

The area of the fenced region is \( 400\sqrt{3} \) square feet.