Math project (calc.) HElppppp?
i got this priject from my instructor and im totally lost. We have to pretend this comapny has hired us for a gutter desgn job…
the letter states….
we are contacting your firm to determine the optimum gutter design for the development.
An optimum gutter will, of course, carry a maximum amount of water, and our design constraints
require that it be manufactured from a 22 inch wide piece of material (which we can order it in the
lengths required to line different roof sections). Due to a machine limitation, we MUST make two folds,
or bends in the width of the metal to form the shape of the gutter. We can make those folds at any
angle needed, however.
We therefore have contacted you to find the gutter design(s) which will carry the most water from the
roof. Because we have already begun the construction of the office buildings, we unfortunately require
a fairly speedy response from you, and therefore MUST have your report by the deadline. We are
not able to give you the perimeter of the roofline, since the final design keeps changing. (Those darn
stonecutters) So is this job even possible? If it is, it would really help if you could give us THREE
different options for the shape of the gutter. It would also be nice to know how far off from optimal we are.
what in the world am i suposed to do??????????
ooo Gerry, you are the best ever…
Essentially you have to take a slice out of that gutter.
That will give you the bottom and two sides of a quadrilateral. The top of the quadrilateral is air. What you have to do is find the quadrilateral(s) that maximize the area and use only those 22 inches for the bottom and sides. What is clear is that the bottom must be parallel to the "flat" ground To see a labeled picture of this problem click on the following link
http://i369.photobucket.com/albums/oo133/gerryrains/Gutter.jpg
The total area of this "not-enclosed" quadrilateral is found by adding the area of the two right triangles to that of the rectangle. To do this we must compute the height of the rectangle.
Note side2 = 22 – (side1 + bottom)
h = cos(a) * side1 = cos(b) * side2
So the total area is:
[(1/2) * (cos(a) * side1) * ((sin(a) * side1)] +
[(bottom) * (cos(a) * side1)] +
[(1/2) * (cos(b) * (22 - (side1 + bottom))]
I will have to finish this tomorrow for you. I am just too sleepy to concentrate. If somebody else answer this question I'll just delete my partial answer.
I'm sorry but I have come down with the flu and can't go on. This turned out to be a real mess, but side1 is equal to side2 and angles a and b are the same.
I just can't go on and am heading back to bed. Look at the drawing and sketch it with symmetrical sides. Compute the height with trig. Notice that two triangles are congruent and that the sum of their area, ((sin(a) / side1) / 2) is therefore bottom * sin(a) / side1 which is equal to side2. This is a trigonometric mess but add those to the size of bottom * h and do a nightmare differentiation. If you extend this, I'll give it a further look when I feel human. It doesn't look like anybody else is eager to deal with it.
:<
.
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Essentially you have to take a slice out of that gutter.
That will give you the bottom and two sides of a quadrilateral. The top of the quadrilateral is air. What you have to do is find the quadrilateral(s) that maximize the area and use only those 22 inches for the bottom and sides. What is clear is that the bottom must be parallel to the "flat" ground To see a labeled picture of this problem click on the following link
http://i369.photobucket.com/albums/oo133/gerryrains/Gutter.jpg
The total area of this "not-enclosed" quadrilateral is found by adding the area of the two right triangles to that of the rectangle. To do this we must compute the height of the rectangle.
Note side2 = 22 – (side1 + bottom)
h = cos(a) * side1 = cos(b) * side2
So the total area is:
[(1/2) * (cos(a) * side1) * ((sin(a) * side1)] +
[(bottom) * (cos(a) * side1)] +
[(1/2) * (cos(b) * (22 - (side1 + bottom))]
I will have to finish this tomorrow for you. I am just too sleepy to concentrate. If somebody else answer this question I'll just delete my partial answer.
I'm sorry but I have come down with the flu and can't go on. This turned out to be a real mess, but side1 is equal to side2 and angles a and b are the same.
I just can't go on and am heading back to bed. Look at the drawing and sketch it with symmetrical sides. Compute the height with trig. Notice that two triangles are congruent and that the sum of their area, ((sin(a) / side1) / 2) is therefore bottom * sin(a) / side1 which is equal to side2. This is a trigonometric mess but add those to the size of bottom * h and do a nightmare differentiation. If you extend this, I'll give it a further look when I feel human. It doesn't look like anybody else is eager to deal with it.
:<
.
References :