Question

A long rectangular sheet of metal, 12 inches wide is to made into a rain gutter by turning up two sides at angles of 120° to the sheet. How many inches should be turned up to give the gutter its greatest capacity?

Answer 1

If the sides of the gutter have length x, then the maximum capacity happens when the cross-section has maximum area. That is a trapezoid.
The depth of the gutter is x sin60°
So, the trapezoid has area a = (12-2x + 2x cos60°)/2 * x sin60°
maximum area is where da/dx = 0
Now take it away. I get x=4

Answer 2

oops
a = (12-2x + 12-2x+2x cos60°)/2 * x sin60°

Answer 3

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Socratic › ... › Solving Optimization Problems
A long rectangular sheet of metal, 12cm wide, is to be made ...

The result is in cm not in inches but it doesn't matter.

Remark:

It is written:
y' = ( 12 − 3 x ) ( √3 / 2 ) = 0

It should be written:
A' = ( 12 − 3 x ) ( √3 / 2 ) = 0

Now multiply both sides by 2 / √3

12 - 3 x = 0

12 = 3x

12 / 3 = x

4 = x

x = 4

Second derivative test:

If A" ( x ) < 0 then function has a local maximum at x.

If A" ( x ) > 0 then function has a local minimum at x.

A" ( x ) = - 3 ∙ √3 / 2 = - 2.598 < 0

Function has a local maximum at x = 4