A rectangular garden must have a perimeter of 155 feet and an area of at least 1400 square feet. Describe the possible lengths of the garden. Round your answers to the nearest foot. The approximate length of the garden is at least feet and at most 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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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.

it is contrapositive

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