Thread: Constructing a function in terms of theta.

1. A rain gutter is to be constructed from a length of sheet metal that is 24cm wide. TO form the gutter the width is folded along its length in three equal sections, so that both ends form an angle theta with the middle section (horizontal). Find the angle that will maximize the capacity of the rain gutter.



I know that you must find the absolute maximum value of theta in the function which would just be taking the first derivative and letting theta = 0 or doesn't exist.


I used one of the triangle identities of 2^2= 3 + 1 to make it into:

8^2=(4sqrt3)^2 + 4^2



However the question does not provide the original function but rather I am supposed to construct in terms of theta. and I have no clue how to do this.

Any help would be greatly appreciated.

I think a good first step would be to draw the completion of the trapezoid. If you do that, the variable theta will show up in another location because of a geometric theorem. Do you see what I'm driving at?

No sorry I don't think I follow. There's some gaps in my basic mathematical knowledge that I'm desperately trying to fill again since I have forgotten them.

Ok. Draw your completed trapezoid as a first step. Now theta is the acute angle between the left side and the horizontal, correct? It will also be equal to the acute angle between the left side and the top line of the trapezoid. This is so because of alternate interior angles. Do you follow me now?

Ah okay I see. But how am i supposed to construct a function? That is finding the cross-sectional area of the trapezoid as a function of theta. How do I manipulate that area of a trapezoid function to have a theta variable?

I think you can assume that the angle theta will be less than a right angle. Because you need to get theta into your area function, I think it would be wise to split the trapezoid into three regions: two triangles and a rectangle. Could you then produce the formula for the sum of the areas of those three regions? Don't worry just yet about getting theta into the formula. Just stick some labels on things and get the area for now.

Area of trapezoid: [(length of base of rectangle) x height] + [(length of base of triangle) x height]

Since there are two equal triangles, division by two is not necessary. So how does theta fit into all of this? Since the height and the base of the triangle is unknown.

Excellent. Now, there are two unknowns in your area formula, right? You've got height, and length of base of triangle. Can you see a way to express those two quantities in terms of theta?

Since sin x = opp/hyp and opp is the height. 8sin x = height. Alternatively cos x = adj/opp and adj is the base of the triangle. Therefore 8cos x = base

Would I be correct in assuming:

Area of trapezoid = 8(8sin x) + (8cos x)(8sin x)
where x = theta.

Looks great! Now what do you need to do next?