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# Optimization Problems: Minima/Maxima

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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 Thumbnails

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 Thumbnails

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.

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