SOLUTION: An architect is designing an atrium for a hotel. The atrium is to be rectangular with a perimeter of 720ft of brass piping. What dimensions will maximize the area of the atrium?

Question 191814: An architect is designing an atrium for a hotel. The atrium is to be rectangular with a perimeter of 720ft of brass piping. What dimensions will maximize the area of the atrium?
Answer by Earlsdon(6294) (Show Source): You can put this solution on YOUR website!
The perimeter of a rectangle is given by:
and this is equal to 720. so...
Divide both sides by 2 to get:
which can be written as:

The area of a rectangle is given by:
Substitute to get:
Simplify.
Now you have a quadratic equation:
When graphed, this will show a parabola opening downwards so there will be a maximum point on the curve and this occurs at the vertex of the parabola. To find the value of W that corresponds to this point, use where a = -1 and b = 360.

So the maximum area is obtained when W = 180ft. and L = 180ft.
This should come as no surprise as it is well-known that the maximum area of a rectangle enclosed by a given perimeter is, in fact a square.
Check:
Substitute L = 180 and W = 180

...the given perimeter.

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