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

Question

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?

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Adalynn 1 week 2021-09-12T23:08:37+00:00 1 Answer 0

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    2021-09-12T23:10:16+00:00

    Answer:

    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

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