Answer:
Step-by-step explanation:
Perimeter of a garden = 2(L+W)
Area of the gerden = LW
L is the length
W is the width
Given
Perimeter = 155ft
Area = 1400ft²
Substitute
1400 = LW
155 = 2L+2W
from 1;
W = 1400/L
Substitute into 2;
155 = 2L + 2(1400/L)
155 = 2L + 2800/L
155L = 2L² + 2800
2L²-155L + 2800 = 0
Factorize
L = 155±√155²-4(2)(2800)/4
L = 155±√24025-22400/4
L = 155±√1625/4
L = L = 155±40.31/4
L = 194.31/4
L = 48.82 feet and;
L = 155-40.31/4
L = 28.67ft
Hence the approximate length of the garden is at least 28.67 feet and at most 48.82 feet.
The possible lengths of the garden are:
The least approximate length = 29 feet
The most approximate length = 49 feet
The area of the rectangle is at least 1400 square feet
That is:
Area ≤ 1400
Let the length be represented by L
Let the width be represented by W
Let the area be represented by A
The area of a rectangle is:
Area = Length x Width
A = LW
LW ≥ 1400....................(1)
The perimeter of the rectangle is 155 feet
P = 2(L + W)
2(L + W) = 155............(2)
Make W the subject of the formula
L + W = 155/2
L + W = 77.5
W = 77.5 - L.............(3)
Substitute W = 77.5 - L into equation (1)
L(77.5 - L) ≥ 1400
77.5L - L² ≥ 1400
-L² ≥ 1400 - 77.5L
0 ≥ L² - 77.5L + 1400
L² - 77.5L + 1400 ≤ 0
Solving the quadratic inequality above
29 ≤ L ≤ 49
The least approximate length of the garden = 29 feet
The most approximate length of the garden = 49 feet
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