Expert: Paul Klarreich - 11/28/2011

Question
A building with a rectangular base is to be constructed on a lot in the form of a right triangle with legs 18m and 24m. if the building has one side along the hypotenuse of the triangle find the dimensions of the base of the building for maximum floor area.

Answer
Questioner: rachel
Country: Manila, Philippines
Category: Calculus
Private: No
Subject: calculus-derivatives
Question: A building with a rectangular base is to be constructed on a lot in the form of a right triangle with legs 18m and 24m. if the building has one side along the hypotenuse of the triangle find the dimensions of the base of the building for maximum floor area.
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I have changed your dimensions to 3 and 4.  Obviously the hypotenuse is 5.  (You can just multiply back by 6 at the end.)

In the diagram,  ABC is your 3-4-5 triangle, and  CDF is similar, as is BDE.

Set  CD = a.  This is going to be our variable.

CD = a = 3a/3
CF = 4a/3
DF = 5a/3

Since CB = 3, and  CD = a,  DB = 3-a

Your rectangle has height h, and width  5a/3.

Using proportions:

DE   BD
-- = --
CF   DF

or:

 h    3 - a
---- = -----
4a/3    5a/3

 h    3 - a
---- = -----
 4      5

Solving for h:

h = 4(3 - a)/5

Now

Area = width * height

Area = 5a/3  * 4(3 - a)/5

Area = a/3 * 4(3 - a)

Area = (4/3)a(3 - a)
Area = (4/3)(3a - a^2)
Now do our stuff.

dArea/da = (4/3)(3 - 2a)

Set that to zero and solve:  a = 3/2

So W = 5(3/2)/3 = 5/2

and h = 4(3 - (3/2))/5 = 4(3/2)/5 = 6/5

Multiply back by your 6, now, and you are done.

Suggestion:  See:

http://en.allexperts.com/q/Calculus-2063/2009/11/Maximum-minimum-problem-41.htm