Question #341dd

1 Answer
Jack
Sep 29, 2016

A square of side 4cm

Explanation:

The rectangle in question will have perimeter #P = 16#
The perimeter of a rectangle is #P = 2h + 2w# where #h# and #w# are the height and width, respectively.

Combining equations, it is clear #16 = 2h+2w => h+w = 8#

It is also useful to note, then that #h = 8-w#

We know the area of the rectangle is #A = h*w#

Substituting in our equation above, we can say:
#A = (8-w)*w# or #A = 8w-w^2#

Now we differentiate area with respect to the width and set the equation equal to #0# (notice that if we solved for height there would be no difference, since the sides are arbitrarily named).

#(dA)/(dw) = 8-2w = 0#

Solving, we see #w = 4#.
Since #8 = w+h#, #h = 4# as well.
Thus, our solution is a square with #w=4# and #h=4#.

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