A community is building a square garden with a walkway around the perimeter with the design shown at the right. Find the side length of the inner square that would make the area of the inner square equal to 75% of the total area of the garden. Round to the nearest tenth of a foot.

1. What is an expression for the area of the inner square?

2. What is the area of the entire garden?

3. What is 75% of the area of the entire garden?

4. Write an equation for the area of the inner square using the expressions from Steps 1 and 3.

5. Solve the quadratic equation. Round to the nearest tenth of a foot.

6. Which solution to the quadratic equation best describes the side length of the inner square? Explain.

17.3 feet

Step-by-step explanation:

The side length of the outer square (the whole garden) is 20 feet and the area of a square is length squared.

20² = 400

For the first question, the area of the inner square will be

X = Length * Width

The area of the entire garden which is also the area of the outer square is

Length * width

= 20 * 20

= 400 feet

For the third question, we were asked to find 75% of the area of the entire garden which is

75/100 * 400

=300 feet

For the fourth question, we are now asked to write an equation of the area of the inner square using steps 1 and 3

x² = 0.75 (20²)

x² = 300

For the fifth question

x² = 0.75 (20) ²

x² = 300

x = √300

x = ( + or-) 17.32 feet

And for the last queston

x = + 17.32 feet since the length shouldn't be a negative

x = 17 ft ... (nearest 10th of a foot)

Step-by-step explanation:

Given:-

- The side length of inner square = x ft

- The side length of outer square = 20 ft

Find:-

Find the side length of the inner square that would make the area of the inner square equal to 75% of the total area of the garden.

Solution:-

- Compute the area of the inner square (A_i) using side length (x):

A_i = side length ^2

A_i = x^2 ft^2

- The area of the entire garden is the area (A_o) of the outer square with side length equal to 20 ft.

A_o = side length ^2

A_o = 20^2

A_o = 400 ft^2

- The 75% of the entire garden is:

(75 / 100) * A_o

(75 / 100) * 400

300 ft^2

- The area of the inner square is 75% of the entire garden the mathematical expression for it is:

A_i = 300 ft^2

x^2 = 300 ft^2

- The side length required for the inner square is the solution to the quadratic equation above:

x = √300

x = 17.320508ft.

x = 17 ft ... (nearest 10th of a foot)