AboutPaul Klarreich Expertise All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions.
I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.
Experience I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.
Expert: Paul Klarreich Date: 6/24/2008 Subject: Maximum-minimum problems
Question Find the longest beam that can go around a hallway with width A to a hallway with a width B.
Tried everything, Obviously involves similar triangles and some trig functions, but I cannot find a significant relationship. Looking to find the MAX length that La + Lb (hypotenuse of the largest triangle) can be through optimization I believe, don't quote me on that. I think the picture is an accurate representation.
Answer Questioner: Willis
Category: Calculus
Private: No
Subject: Optimization
Question: Find the longest beam that can go around a hallway with width A to a hallway with a width B.
Tried everything, Obviously involves similar triangles and some trig functions, but I cannot find a significant relationship. Looking to find the MAX length that La + Lb (hypotenuse of the largest triangle) can be through optimization I believe, don't quote me on that. I think the picture is an accurate representation
.............................................
Hi, Willis,
>> What picture?
Ah, well, it probably looks like this:
(I remember this from some years ago. Alas, I cannot find the book that had it -- it's probably the old Thomas book from the 1950's.)
There probably is an easier way, but this is the best I can come up with.
You have a line that passes through the point (a,b) and has an x-intercept, which we will call, er,.. x0.
(x0 - a)
Its slope is ---------
(0 - b)
(x0 - a)
= ---------
- b
a - x0
= ------
b
a - x0
Its equation is y - b = --------(x - a)
b
a - x0
y - b = --------(x - a) + b
b
(a - x0)(x - a) + b^2
y = ---------------------
b
Now the y-intercept is:
(a - x0)(0 - a) + b^2
y0 = ---------------------
b
(a - x0)(- a) + b^2
y0 = ---------------------
b
-a^2 + a x0 + b^2
y0 = ---------------------
b
Now the length of the segment is x0^2 + y0^2 [the length^2, actually]
and we have to MINIMIZE this. (Yes, minimize, because if the beam is longer than the minimum, it won't go around.)
