Explanation
From the statement, we must find the dimensions of a rectangle with:
• perimeter P = 194,
,
• the largest area.
We consider a rectangle with sides x and y. We will write formulas for P (the perimeter) and A (the area). Then we will express the area in terms of one side A(x), and maximize the function.
(1) The perimeter of the rectangle is:
[tex]P=2(x+y)\rightarrow y=\frac{P}{2}-x=\frac{194}{2}-x=97-x.[/tex]
(2) The area of the rectangle is:
[tex]A=x\cdot y=x\cdot(97-x)=-x^2+97x.[/tex]
(3) To maximize the area A(x), we compute and make equal to zero its first derivative, then we solve for x:
[tex]A^{\prime}(x)=-2x+97=0\Rightarrow2x=97\Rightarrow x=\frac{97}{2}.[/tex]
(4) From point (1), we find the length of the other side:
[tex]y=97-x=97-\frac{97}{2}=\frac{97}{2}.[/tex]Answer
The dimensions of the rectangle with a perimeter of 194 and the largest area are:
[tex]\frac{97}{2}\text{ and }\frac{97}{2}[/tex]
