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
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
- 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 ...