phyllis wants to fence in a rectangular garden with maximum area. she has 22 feet of fence. all the lengths of the sides of her garden must be whole numbers of feet. what is the best choice available to phyllis for the side lengths of her garden? Why?
The maximum area will be the closest to a square.
How about 5 by 6?
With 5 it would be 25 ft needed and with 6 it would be 0 but she only has 22.
I said the closest to a square -- which would be 5 feet by 6 feet.
P = 2L + 2W
22 = 2(5) + 2(6)
22 = 10 + 12
The area would be 30 square feet.
To find the best choice for the side lengths of Phyllis's garden, we can consider the relationship between the sides of a rectangle and its area. Let's denote the length of one side as "x" and the length of the adjacent side as "y."
A rectangle's area is given by the formula: area = length × width.
Since Phyllis wants to maximize the area, we need to maximize the product of the side lengths. However, we have a constraint that the perimeter (sum of the four sides) should be 22 feet.
Let's try different values of x and calculate the corresponding value of y based on the perimeter constraint:
1. x = 1 foot:
In this case, the perimeter becomes: 1 + y + 1 + y = 2 + 2y.
Since 2 + 2y = 22, solving for y, we get y = 10. Therefore, the sides would be 1 foot by 10 feet, and the area would be 1 × 10 = 10 square feet.
2. x = 2 feet:
The perimeter becomes: 2 + y + 2 + y = 4 + 2y.
Solving 4 + 2y = 22, we find y = 9. The sides would be 2 feet by 9 feet, and the area would be 2 × 9 = 18 square feet.
3. x = 3 feet:
The perimeter becomes: 3 + y + 3 + y = 6 + 2y.
Solving 6 + 2y = 22, we get y = 8. The sides would be 3 feet by 8 feet, and the area would be 3 × 8 = 24 square feet.
4. x = 4 feet:
The perimeter becomes: 4 + y + 4 + y = 8 + 2y.
Solving 8 + 2y = 22, we find y = 7. The sides would be 4 feet by 7 feet, and the area would be 4 × 7 = 28 square feet.
By comparing the areas, we see that the greatest area (28 square feet) is achieved when Phyllis chooses side lengths of 4 feet by 7 feet. Therefore, the best choice available to Phyllis for the side lengths of her garden is 4 feet by 7 feet.