A rain gutter is to be made of aluminum sheets that are 12 inches wide turning up the edges 90 degrees. What depth will provide maximum cross-sectional area and hence allow the most water to flow?
I've attached a sketch of the gutter.
1. The area A of the cross section is calculated by:
A = x * y
2. You already know that x + y + x = 12 ==> y = 12 - 2x
3. Plug in this term for y into the first equation and you'll get a function with respect to x:
A(x) = x * (12 - 2x) = 12x - 2x²
4. Now use the first derivative to get the value of x which yields an extreme area:
I've got x = 3. Therefore y = 6 and the maxiumum area is A_{max} = 18
I've attached the protocol of generating the sketch of the gutter .
1. Choose a thicker line. Draw a rectangle.
2. Erase the top-line of the rectangle. Use the magnifying glass of paint to work accurately!
3. Copy the topless rectangle and paste it above the original rectangle, shifted to the right.
4. Connect the vertices of the 2 rectangles. Since both shapes are congruent the connecting lines are parallel automatically.
5. Erase the invisible part of the rear line.
6. Close the front rectangle with a thin top-line and fill the area by light grey color.
7. Sign your artwork and paste a $500 price-tag on it.