Question

a gym consists of a rectangular region with a semi-circle on each end. if the perimeter of the room is to be a 200 m running track, find the dimensions that will make the area of the rectangular region as large as possible.

Answer 1

The dimensions of the rectangle are:

l = 50 m

b = $100/\pi$ m

What is a perimeter in math?

The perimeter is the length of the outline of a shape. To find the perimeter of a rectangle or square you have to add the lengths of all the four sides.

How do we find a perimeter of a rectangle?

The perimeter of a rectangle,denoted by P is given by the formula, P=2l+2b, where l is the length and b is the breadth of the rectangle.

Given:

As per the question:

Perimeter of the room is given as P = 200 m

The region is rectangular having a semicircle at each end.

Now,

Let 'l' be the length of the rectangle, 'b' be its breadth and 'r' be the radius of the semi-circle at each end.

Then, Area of the given rectangle, A = lb

Perimeter of the room, P is =$\pi r+l+\pi r+l=2\pi r+2l=\pi b+2l$

Therefore,  $\pi b+2l=200$

$b=(200-2l)/\pi$

Now,

Area, A = $l(200-2l)/\pi=(200l-2l^{2} )/\pi$

Now, differentiate A w.r.t l:

Again differentiating w.r.t 'l', we get:

$d^{2} A/dl^{2} =-4l/\pi$< 0

Thus we get maximum are when $dA/dl=0$

Therefore,

$(200-4l)/\pi=0$

l = 50 m

Now, from

$\pi b+2l=200$

$\pi b=200-2*50$

$b=100/\pi$

$r=b/2=50/\pi$

To know more about area of a recatangle, visit the link

brainly.com/question/20693059

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