Question

A rectangular field is to be fenced on three sides with 1000 m of fencing. The fourth side is a straight river's edge that will not be fenced. Find the dimensions of the field so that the area of the enclosure is 120000 square meters.

Answer 1

Answer:

$300m\times 400m$ or$200m\times 600m$

Step-by-step explanation:

We are given that

Fencing used on three sides=1000 m

Area of field enclosure=120000 square meters

Let x be the length and y be the width of rectangle.

Fencing used on three sides=2x+y

$2x+y=1000$

$y=1000-2x$

Area of  field=$xy=x(1000-2x)$

$-2x^2+1000x=120000$

$-x^2+500x=60000$

$x^2-500x+60000=0$

Using quadratic formula

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{500\pm\sqrt{(500)^2-4(60000)}}{2}$

$x=\frac{500\pm 100}{2}$

$x=\frac{500+100}{2}=300m$

$x=\frac{500-100}{2}=200 m$

$y=1000-2(300)=400 m$

$y=1000-2(200)=600m$

Dimension of the field

$300m\times 400m$ or$200m\times 600m$

Answer 2

Answer:

length = 500 m , width = 250 m

Step-by-step explanation:

Let the length of the field is L and the width is W.

Area of the field , A = 120000 sq. metre

Length of fencing = 1000 m

Total length of fence = L + 2 W = 1000

L = 1000 - 2 W

Area = Length x width

A = L x W

A = (1000 - 2W) x W

A = 1000 W - 2 W²

for mxima and minima,

dA/dW = 0

1000 - 4W = 0

W = 250 m

L = 1000 - 2 x 250 = 500 m

Thus, the length of the field is 500 m and the width of the field is 250 m.