Arectangle of perimeter 100 units has the dimensions shown. its area is given by the function a = w(50 - w). what is the greatest area such a rectangle can have?

the rectangle shown has 50-w on the top and bottom.
and a w on its left side and right side.

The greatest area of rectangle is:

625 square units.

Step-by-step explanation:

It is given that:

A rectangle of perimeter 100 units has the dimensions as:

50-w on the top and bottom.

and a w on its left side and right side.

i.e. we may say the length of the rectangle is:

50-w

and the width of the rectangle is:

w

Now, we need to find the greatest area of rectangle.

As the area of rectangle is:

A = w(50 - w)=50w-w^2

Now, to find the maximum area we differentiate the Area with respect to the width as:

Hence, to obtain the maximum area the width of the rectangle is: 25 units.

and that of the length of the rectangle is:

50-25=25 units.

Hence, the dimensions of rectangle in order to obtain the maximum area is:

25 units by 25 units.

So, the area of rectangle is:

Hence, the greatest area of rectangle is:

625 square units.