Question

find the largest area for a rectangle if you have 2896 feet

Answer 1

2896 feet is the perimeter?

Answer 2

how do you find the area

Answer 3

First a picture :constraints :\[perimeter = 2896 = 2\times h + 2 \times w\]\[area = A = h \times w\]so\[h + w = 1448\]\[h=1448-w\]thus\[A = (1448-w)w= 1448w - w^{2}\]that is : the requirement on the perimeter allowed us to express one side's length in terms of the other, and that has simplified the expression for the area. Plot the curve of this quadratic, it is concave downwards and the area is zero if w = 0 or w = 1448so when does the curve hit its pea k?

Since this is a differential equations board I suppose we'd better use calculus, although good old completing the square would do it nicely also.\[\frac{dA}{dw}=1448-2w\]set that to zero for the peak :\[1448=2w\]\[w=724\]leaving\[h=1448 - w=724\]a square that is ( that's generally true as the problem is phrased ), giving :\[A=724^{2}=524176\]square feet.