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Gutter Bends

April 19th, 2011 |

A rain gutter is to be constructed from a metal sheet (spreadsheet)?

A rain gutter is to be constructed from a sheet metal width 30 cm folding third leaf on each side through an angle θ. How θ must be such that the channel will carry the maximum amount water? here is the link for the photo of the problem http://www.webassign.net/scalc/4-6-38.gif

We need to start building a function that gives the area of the channel in terms of theta. To calculate the area, dividing the channel cross section into two triangles and a rectangle. Then is a simple ASCII art diagram of how to divide the way, I hope you can understand (just one of the triangles are drawn for simplicity, the other is the mirror image) ………… ….. ……………__ w ………….|…./. …… ……|…/ 10 .. …….. h. | .. / …. ……….|./.. …. θ ____|/___ .. ….. 10 The ditch area therefore equals the area of rectangle the area of triangles. A = triangle + 2 * = 10h + 2 * (1 / 2 hw) Using trigonometry working out w. h h = 10 w = 10 cos θ sin θ A = 10h + + hw = 100sinθ 100sinθcosθ Differentiating the expression (or use product rule for sin.theta cos.theta, or write 100sinθcosθ as 50sin2θ using the double angle formula) + A = 100sinθ 50sin2θ dA / dt = 100cosθ + 100cos2θ A to a maximum or minimum the derivative must be zero 100cosθ + = 0 + cos.theta 100cos2θ cos2θ = 0 Use the double angle formula cos2θ write as 2cos ^ 2 θ – 1 cos.theta + 2cos ^ 2 θ – 1 = 0 2cos ^ 2 θ + cos.theta – 1 = 0 This quadratic factorization (2cosθ – 1) (cos.theta + 1) = 0 This means that cos θ = 1 / 2 or cos θ = 1, stating: θ = pi / 3 or θ = pi This means that to achieve the maximum area should be θ pi / 3 = 60 degrees You can verify that this is the best by replacing these values of theta and the two end points (theta = 0, theta = pi), and seeing that pi / 3 gives the maximum. The maximum is therefore A = + 100sinθ 100sin 100sinθcosθ = (pi / 3) + 100sin (pi / 3) cos (pi / 3) = 100 sqrt (3) / 2 + 100 sqrt (3) / 2 * 1 / 2 = 100 * 2.3 * sqrt (3) / 2 = 3 * 25 * sqrt (3) = 75 sqrt (3) = 129.9 … = 130 cm ^ 3

FAIL! Riley crashes into a building and bends the Rain Gutter with the Lawn Mower!!

Malco S9 9-Inch Aluminum Hand Seamer

$48.60

For making clean and accurate angle bends for aluminum roof flashing or fascia corners. Jaws depth graduations in 1/4″ increments to a maximum 2″ bending depth. Lightweight die cast aluminum jaws with compound handle leverage easy to use on-the-job, evenwhen working from heights. NOT WARRANTED FOR USE AS A STRIKING TOOL OR FOR USE ON FERROUS SHEET METAL….

Amerimax Home Products 4′ White Pvc Solid Gutter Cover 85320

$3.55

The gutter cover is designed with 2 flexible beads that bend to fit up to a 14 12 pitch roof. Water flows around the nose of the solid gutter cover and into the gutter through the slotted channel. It will handle the heaviest of rainfall. Helps keep leaves, pine needles, and other debris out of the gutter. Made of a rigid PVC. Easy to install. Starting at one end of the gutter, slightly lift the sh…

MALCO PRODUCTS 5 In. Hand Seamer

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HAND SEAMER W/DIE CAST ZINC JAWS – Ideal for making clean and accurate angle bends for aluminum roof flashing, coil stock or fascia. Jaws feature depth graduations in 1/4 increments to a maximum 2 bending depth. Lightweight die cast zinc jaws with compound handle leverage offer comfort and ease of use on the job, even when working from heights. Durable construction offers long, dependable service …