You are Here: Home >< Maths

# Optimization Problems: Minima/Maxima Watch

1. An arch of a bridge is designed as a rectangle surmounted by a semicircle. If the area of the archway is 100 m^2, determine the radius of the semicircle such that the perimeter of the edge has a minimum value.
I can't solve this.

From the info given I can determine what r would be of the archway. Which can only be one value if I am not mistaken since this is simple geometry. Using (pi *r^2)/2

The perimeter will then be 4r, which will provide a differential of 4, which cannot be set to 0 to get a critical point. I'm also assuming the bottom edge is not counted, but it doesn't matter.

Would the 'area of the archway' refer to the entire rectangle or just the circle as I assumed.

Hate havign my math ability being limited by my english ability. See attachment.
Attached Images

2. (Original post by chrislpp)
...
Your diagram is incorrect. You've gone with what you think the arch of a bridge should look like, rather than what the question actually says.

Think of an old railway viaduct, or just "rectangle surmounted by a semicircle"
3. Here is the new diagram.

Perimeter = 2πr + 2r + 2x

Area 100m^2 = (πr^2) / 2 + 2rx

Still don't think this is ready for the derivative tests.
Attached Images

4. (Original post by chrislpp)
Here is the new diagram.

Perimeter = 2πr + 2r + 2x

Area 100m^2 = (πr^2) / 2 + 2rx

Still don't think this is ready for the derivative tests.
2*pi*r is the perimeter of a circle, not a semi-circle.

And why not?
5. Since you know the area is 100, you can use the 2nd equation to write x in terms of r. Then sub into the first equation to get a formula for the perimeter that only involves r. Then differentiate.

Updated: May 21, 2012
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Today on TSR

### Stuck for things to do this summer?

Come and get some inspiration.

Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams