A farmer has 1200 feet of fencing. He wants to enclose a rectangular field bordering a river, WITH NO FENCING NEEDED ALONG THE RIVER. Find the dimensions of the field if the area of the field is 180,000 ft^2. No guess and check. I need an equation.
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Define variables:
Let L denote the length (in feet) of the rectangular field.
Let W denote the width (in feet) of the rectangular field.
Area of a rectangle = length × width. So we have
LW = 180000
Let's assume that the river forms the boundary parallel to the length of the field. Then the fencing must provide the two widths and one length of the field. Hence,
L + 2W = 1200
So we have the system
L + 2W = 1200
LW = 180000
Solve the first equation for, say, L in terms of W:
L = 1200 - 2W
Substitute this for L in the area equation:
(1200 - 2W)W = 180000
1200W - 2W² = 180000
0 = 180000 - 1200W + 2W²
0 = 90000 - 600W + W²
0 = (300 - W)²
0 = 300 - W
W = 300 ft.
L = 1200 - 2W = 600 ft.
Let L denote the length (in feet) of the rectangular field.
Let W denote the width (in feet) of the rectangular field.
Area of a rectangle = length × width. So we have
LW = 180000
Let's assume that the river forms the boundary parallel to the length of the field. Then the fencing must provide the two widths and one length of the field. Hence,
L + 2W = 1200
So we have the system
L + 2W = 1200
LW = 180000
Solve the first equation for, say, L in terms of W:
L = 1200 - 2W
Substitute this for L in the area equation:
(1200 - 2W)W = 180000
1200W - 2W² = 180000
0 = 180000 - 1200W + 2W²
0 = 90000 - 600W + W²
0 = (300 - W)²
0 = 300 - W
W = 300 ft.
L = 1200 - 2W = 600 ft.