Expert: Paul Klarreich - 10/14/2008

Question
Yes sorry the 7.5cm,15cm,7.5 are    7.5cm____/7.5cm
                                          15cm
For a gutter with the sides of 7.5cm bent at an angle, calculate theta
to find the maximum volume of flow through the gutter.
Thanks

Answer
Questioner:   Larissa
Category:  Calculus
Private:  No
 
Subject:  The gutter problem
Question:  Yes sorry the 7.5cm,15cm,7.5 are    7.5cm____/7.5cm
                                         15cm
For a gutter with the sides of 7.5cm bent at an angle, calculate theta
to find the maximum volume of flow through the gutter.
Thanks
...................................
Hi Larissa,

Is this your first attempt at Max-min problems?  If so, the scheme is something like this:

Oh, wait -- I said that already.  But it applies.

You want to maximize the cross-section are a, meaning the area of the trapezoid.  So here

is what you will do:

1. You will draw a diagram.

2. You will determine the height and OTHER base of the trapezoid, based on the angle theta

that you see in this picture (it's just labeled 't')
  15 + 2x
--------------+
\           / |
\      7.5/  | h
 \       /t  |
  -------....+
     15    x

Use your trig.
h = 7.5 sin t
x = 7.5 cos t

A = h/2(b1 + b2)  << area formula for a trapezoid

A = 7.5 sin t(15 + 15 + 2(7.5 cos t))/2

A = 7.5 sin t(30 + 15 cos t)/2

A = (7.5*15/2) sin t(2 + cos t)

Now ignore the constant and take:

A* = sin t(2 + cos t)

to be maxed:

A' = sin t(-sin t) + cos t(2 + cos t)

A' = - sin^2 t + 2 cos t + cos^2 t

make all cosines: Remember: sin^2 = 1 - cos^2

A' = cos^2 - 1 + 2 cos t + cos^2 t

A' = 2 cos^2 t  + 2 cos t - 1

Solve for cos t using the quadratic formula:

       -2 +- sqrt(4 + 4)
cos t = -----------------
             4

       -2 +- 2 sqrt(2)
cos t = -----------------
             4

       -1 +- sqrt(2)
cos t = -----------------
             2

You want the positive root, because  t must be in [0,pi/2]

So: just compute
       -1 + sqrt(2)
cos t = -------------
             2

and plug that into your calculator.  You can do the rest.