Question

of all rectangles with a perimeter of 13​, which one has the maximum​ area? (Give the​ dimensions.) The rectangle that has the maximum area has length ___ and width ___

Answer 1

Let us consider an rectangle of perimiter 13 with length x and width 6.5-x .

Now the area of the rectangle is a=x(6.5-x)=6.5x-x2​​​​​.

Now we need to maximize a. Hence differentiate a with respect to x and equating it to 0 we get,

da/dx=0

=> 6.5-2x=0

=> x=3.25

And d2a/dx2​​​​​​=-2<0

Hence it attain its maximum at x=3.25

Hence the length of the rectangle is 3.25 and the width of the rectangle is 3.25.

And the dimention of the rectangle is 1*1 i.e. It will be an square.