5) A landscaper wants to make a small planter and surround it with edging material for a client. She is deciding between two designs.

Design 1: A square planter
Design 2: A circular planter
Edging material costs $4 per ft. for the straight variety which she would use for Design 1 and $5 per foot for the flexible variety which she would use for Design 2.

a. If she decides on a perimeter of 24 ft. Which design would give her the larger planting area?
b. If she decides to spend $120 on edging material, which Design would give her the larger planting area?

(a) a circle always gives the largest area for a given perimeter.

Among rectangles, a square gives the largest area.

(b)
1: p = 120/4 = 30 ft
area: 7.5^2 = 56.25
2: c = 120/5 = 24
area: = 45.8
So, the expensive fencing reduces the circular area for planting.

To determine which design would give the landscaper a larger planting area, we can calculate the areas of each design and compare them.

a. For Design 1 - Square Planter:
Since the perimeter is 24 ft, each side of the square planter would be 24 ft / 4 = 6 ft.
The area of a square is calculated by multiplying the length of one side by itself: Area = side * side.
So, the area of Design 1 would be 6 ft * 6 ft = 36 sq ft.

For Design 2 - Circular Planter:
The perimeter of the circular planter would be the circumference of the circle, which is equal to 24 ft.
Using the formula for the circumference of a circle: Circumference = 2 * pi * radius,
we can calculate the radius of the circular planter.
24 ft = 2 * π * r
Dividing both sides by 2π, we get r = 24 ft / (2 * π) ≈ 3.82 ft.
The area of a circle is calculated by multiplying π (pi) by the radius squared: Area = π * r^2.
So, the area of Design 2 would be π * (3.82 ft)^2 ≈ 45.65 sq ft.

Therefore, Design 2 - Circular Planter would give the landscaper a larger planting area.

b. To determine which design would give the landscaper a larger planting area based on a budget of $120 on edging material, we need to calculate the length of edging material required for each design and see if it fits within the budget.

For Design 1 - Square Planter:
The perimeter of the square planter would be 24 ft, as given.
Since the edging material costs $4 per foot for the straight variety, the cost of edging material for Design 1 would be 24 ft * $4/ft = $96.

For Design 2 - Circular Planter:
The perimeter of the circular planter would be 24 ft, as given.
Since the edging material costs $5 per foot for the flexible variety, the cost of edging material for Design 2 would be 24 ft * $5/ft = $120.

Since the budget is $120, both designs fit within the budget.

Therefore, we cannot determine which design would give the landscaper a larger planting area based on the budget of $120 on edging material alone.

To determine which design would give the landscaper a larger planting area, we need to compare the areas of each design based on certain constraints.

a. To determine which design would have a larger planting area for a given perimeter of 24 ft, we need to calculate the side length of the square planter and the radius of the circular planter.

Design 1: Square Planter
For a square planter, the perimeter is equal to four times the length of one side. Therefore, if the perimeter is 24 ft, each side would be 24 ft / 4 = 6 ft.

Design 2: Circular Planter
For a circular planter, the perimeter is equal to the circumference of the circle, which is 2πr (where r is the radius). Given that the perimeter is 24 ft, we can calculate the radius as follows:
24 ft = 2πr
r = 24 ft / (2π)
r ≈ 24 ft / 6.28 ≈ 3.82 ft

Now that we have the side length of the square planter (6 ft) and the radius of the circular planter (approximately 3.82 ft), we can calculate the areas of each design to determine which design has the larger planting area.

Design 1: Square Planter
The area of a square is calculated by multiplying the length of one side by itself. Therefore, the area of the square planter is 6 ft * 6 ft = 36 sq ft.

Design 2: Circular Planter
The area of a circle is calculated using the formula A = πr^2. Therefore, the area of the circular planter is π * (3.82 ft)^2 ≈ 45.95 sq ft.

Comparing the areas of the two designs, we find that Design 2, the circular planter, has a larger planting area of approximately 45.95 sq ft. Therefore, if the landscaper wants a perimeter of 24 ft, Design 2 would give her the larger planting area.

b. To determine which design would have a larger planting area for a budget of $120 on edging material, we need to calculate the length of edging material for each design within the given budget.

Design 1: Square Planter
The perimeter of the square planter is equal to four times the length of one side. Therefore, for a given budget of $120 on edging material, the length of edging material for the square planter would be $120 / $4 per ft = 30 ft.

Design 2: Circular Planter
The perimeter of the circular planter is equal to the circumference of the circle, which is 2πr. Therefore, for a given budget of $120 on edging material, the length of edging material for the circular planter would be $120 / $5 per ft = 24 ft.

Now that we have the length of edging material for each design within the given budget, we can calculate the areas of each design to determine which one has the larger planting area.

Design 1: Square Planter
Given that the length of edging material is 30 ft, we can calculate the side length of the square planter as follows:
Perimeter = 4 * Side Length
30 ft = 4 * Side Length
Side Length = 30 ft / 4 = 7.5 ft
Therefore, the area of the square planter would be 7.5 ft * 7.5 ft = 56.25 sq ft.

Design 2: Circular Planter
Given that the length of edging material is 24 ft, we can calculate the radius of the circular planter as follows:
Perimeter = 2πr
24 ft = 2πr
r = 24 ft / (2π) ≈ 3.82 ft
Therefore, the area of the circular planter would be π * (3.82 ft)^2 ≈ 45.95 sq ft.

Comparing the areas of the two designs, we find that Design 1, the square planter, has a larger planting area of approximately 56.25 sq ft. Therefore, if the landscaper wants to spend $120 on edging material, Design 1 would give her the larger planting area.