Arectangular garden must have a perimeter of 150 feet and an area of at least 1200 square feet. describe the possible lengths of the garden. round your answers to the nearest foot.

The length of the garden is: l ≤ 23 ft or l ≥ 52ft

Let the length of the garden be represented by l

Let the width of the garden be represented by w

Perimeter, P =  150 feet

The perimeter of the rectangle  =  2(l  +  w)

2(l  +  w)   =   150

l   +  w    =  75(1)  

The area of the rectangle(A) = Length  x  width

A   =   lw

lw    ≤  1200(2)

Make w the subject of the formula in equation (1)

w   =  75  -  l(3)

Substitute equation (3) into equation (2)

(75 - l)(l)   ≤    1200

75l  -  l²  ≤   1200

0   ≤  l²  -  75l  +  1200

l²  -  75l  +  1200  ≥  0

Solving the quadratic inequality above:

l  ≥  51.86  or  l  ≤  23.13

Therefore, the length of the garden is: l ≤ 23 ft or l ≥ 52

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51.86 and 23.14

Step-by-step explanation:

The rectangular garden must have a perimeter of 150 feet

Perimeter of rectangular garden =

2l + 2w=150 1

The rectangular garden must have a perimeter of 150 feet and an area of at least 1200 square feet.

Area of rectangular garden =

l×w= 1200 feet^2 2

From equation 2, l=1200/w

Put l=1200/w in equation 1

2× 1200/w + 2w = 150

(2400/w) +2w = 150

(2400+2w^2)/w =150

2400+2w^2= 150w

2w^2- 150w+2400=0

Using the general formula

w = [-b+-√(b^2-4ac)]/2a

a = 2, b =-150, c=2400

w =[--150+/-√(-150^2-4×2×2400)]/2×2

=[150+/-√(22500-19200)]/4

=[150+/-√3300)]/4

=(150+57.45)/4 or (150-57.45)/4

w= 207.45/4 or 92.55/4

w= 51.86 or w= 23.14

l = 1200/51.86 or l= 1200/23.14

l = 23.14. or l= 51.86

For an area of at least 1200ft^2

The dimensions are 51.86 and 23.14

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step-by-step explanation: