Posted by **sunny** on Wednesday, May 8, 2013 at 1:03am.

Given that a sports arena will have a 1400 meter perimeter and will have semi-

circles at the ends with a possible rectangular area between the semi-circles, determine

the dimensions of the rectangle and semi-circles that will maximize the total area.

- math -
**Reiny**, Wednesday, May 8, 2013 at 6:54am
Let the radius of the end sem-circles be r

making the width of the rectangle to be 2r

let the length of the rectangle be x

The perimeter of the field is 2x + 2πr

= 1400

x + πr = 700

x = 700-πr

area = πr^2 + 2rx

= πr^2 + 2r(700-πr)

= πr^2 + 1400r - 2πr^2

d(area)/dr = 2πr + 1400 - 4πr = 0 for a max of area

1400 - 2πr = 0

2πr = 1400

r = 700/π

then x = 700 - π(700/π) = 0

Unexpected strange result!

BUT there is nothing wrong with the calculations, it is the wording of the question that is flawed.

Of course the largest area of any region with a given perimeter is a circle, which my solution has shown.

## Answer this Question

## Related Questions

- Geometry - I have a picture of a window showing a square in the center with ...
- calc - Given that a window entails a rectangle capped by a semi-circle, given ...
- Math - A norman window is constructed by adjoining a semicircle to the top of an...
- math - An oval track is made by erecting semi circles on each end of a 58 m by ...
- maths - This taxi sign is 2.1 metres wide and 39cm tall. Assuming the ends are ...
- Math - The first question is this: Helen designs a rectangle with an area of 225...
- mathematics,physics,life science,LO,afrikaans - A number of circles are given ...
- Maths - the figure below shows 3 similar circles that just touch one another.A,...
- math - Six unit circles are arranged inside a rectangle. the circles are tangent...
- calculus - A Norman window has the shape of a rectangle surmounted by a ...