The owner of the Rancho Grande has 2,980 yd of fencing with which to enclose a rectangular piece of grazing land situated along the straight portion of a river. If fencing is not required along the river, what are the dimensions (in yd) of the largest area he can enclose?

rectangle with maximum area has dimensions of 745 yd x 1490 yd

Step-by-step explanation:

the rectangular area is

Area = x*y, where x = side along the river, y = side perpendicular to the river

since we have only 2980 yd of fencing, the total fencing (perimeter) will be

x+2*y = 2980 yd = a

then solving for x

x = a - 2*y

replacing in the area expression

A=Area = x*y = (a - 2*y) * y = a*y - 2*y²

the maximum area is found when the derivative with respect to y is 0, then

dA/dy = a - 4*y = 0 → y=a/4 = 2980 yd / 4 = 745 yd

then

x = a - 2*y = a - 2 * a/4 = a/2 = 2980 yd / 2 = 1490 yd

then the rectangle with maximum area has dimensions of 745 yd x 1490 yd