# Maximize Area

joemama69

Note the picture

## The Attempt at a Solution

A of both trianles = xsinQ$$\sqrt{x^2 - x^2(sinQ})^2$$

A of rectangle = (w-2x)(xsinQ)

Total A = (w-2x)(xsinQ) + xsinQ$$\sqrt{x^2 - x^2(sinQ)^2}$$

A = wxsinQ + 2x2sinQ + x2sinQcosQ

Ax = wsinQ + 4xsinQ + 2xsinQcosQ = 0

-wsinQ = x(4sinQ + 2sinQcosQ)

x = -w/(4 + 2cosQ)

AQ = wxcosQ + 2x2cosQ + x2(sinQ)2 - x2(cosQ)2 = 0

It gets pretty messy when i plug in x, are there anny errors so far

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joemama69
anybody, can i get some help

Mentor

Note the picture

## The Attempt at a Solution

A of both trianles = xsinQ$$\sqrt{x^2 - x^2(sinQ})^2$$

A of rectangle = (w-2x)(xsinQ)

Total A = (w-2x)(xsinQ) + xsinQ$$\sqrt{x^2 - x^2(sinQ)^2}$$

A = wxsinQ + 2x2sinQ + x2sinQcosQ

Ax = wsinQ + 4xsinQ + 2xsinQcosQ = 0

-wsinQ = x(4sinQ + 2sinQcosQ)

x = -w/(4 + 2cosQ)

AQ = wxcosQ + 2x2cosQ + x2(sinQ)2 - x2(cosQ)2 = 0

It gets pretty messy when i plug in x, are there anny errors so far
What's the problem you are trying to solve? The picture doesn't give me any idea of what you're trying to do.

joemama69
you must Maximize the area, note the title

Mentor
Some context would have been nice, such as that the area is the cross-sectional area of what looks to be a trough, formed from sheet metal that is W units wide.

You have a sign error in this line:
A = wxsinQ + 2x2sinQ + x2sinQcosQ
The first '+' sign should be '-'. That makes Ax = WxsinQ - 4xsinQ + 2xsinQcosQ = sinQ[W -4x + 2xcosQ].

The sign error also affects AQ, which I get as xWcosQ -2x2cosQ + x2cos2Q - x2sin2Q.
About the only thing I can think of to do is to factor x out of each term in this partial.

The critical points are where Q = 0 or when x = 0, either of which gives you an area of 0, and whatever you get from some messy equations that remain. One of these is W - 4x + 2xcosQ = 0, which you can solve for x in terms of Q (and W), or solve for Q in terms of x (and W). Whichever variable you solve for, substitute that in the other equation that comes from AQ.

joemama69
Ya, this is pretty much where i ran into problems, i pluged in my x = w/(4-2cosQ) into A sub Q

AQ = 0 = (w2cos)/(4-2cos) - 2w2cos/(4-2cos)2 + (2w2cos2)/(4-2cos)2 - (w2sin2)/(4-2cos2)

I would have to solve that for Q (i got lazy and didnt type them, just assume they are there), which means (tell me if I am wrong), i only have to solve the numerators of the expression for 0 in terms of w

I got Q = 0 & 2pi which would make the gutter flat, what happened

Mentor
I didn't work it through that far, so I'll assume that the work you have in post 6 is correct. You can't just "solve the numerators," as you put it, since not all of your expressions have the same denominator; the first has a denominator of 4 - 2cosQ. You'll need to multiply that term by (4 - 2cosQ)/(4 - 2cosQ), and then you can factor out 1/(4 - 2cosQ)2 from all four terms. Finally, you'll need to see what values of Q (theta) make AQ equal to 0.

As I said before, Q = 0 and x = 0 are critical values, but they produce a trough with cross-sectional area 0, so they are not the values you want.

joemama69
ok i was able to factor out the w^{2} and the (4-2cos)^{2} and I got

4cos - 2cos^{2} - 2cos + cos^{2} - sin^{2} = 2cos - con^{2} - sin^{2} = 0

I got Q = pi/3, 5pi/3, 7pi/3

how do i calc which one is the maximum, there are many possibilities

Mentor
Due to the fact that you're working with a physical object, a trough, there is a natural range of values for Q (theta). That should help you eliminate many of the possibilities. If Axx < 0 and AQQ < 0, you're at a local maximum.