# Thread: Maxima and Minima Problems

1. ## Maxima and Minima Problems

The strenghth of a wooden beam with a rectangular cross section and fixed length is proportional to the product of its width and the square of its height where the proportionailty constant is 2 psi.

Suppose the roofing company wants to make wooden beams of fixed length with rectangular cross sections from logs that are 4 inches in diameter. If the height of each beam must be at least 3.5 inches find the dimesnsions of the strongest beams.

2. I honestly have no clue how to do this problem

this is a maximum problem right?

i dont even know where to start, the wording is confusing

3. bump

4. really need this one guys

5. Originally Posted by pedromartinez
The strenghth of a wooden beam with a rectangular cross section and fixed length is proportional to the product of its width and the square of its height where the proportionailty constant is 2 psi.

Suppose the roofing company wants to make wooden beams of fixed length with rectangular cross sections from logs that are 4 inches in diameter. If the height of each beam must be at least 3.5 inches find the dimesnsions of the strongest beams.
Start by drawing a picture. You can take the cross section of a log to be the circle around the origin with radius 2: $x^2+ y^2= 4$. The rectangular cross section of a beam made from that log will be a rectangle with vertices at (x,y), (-x,y), (-x,-y), and (x, -y) where x and y satisfy $x^2+y^2= 4$. Now, if "w" is the width of the rectangle, x= w/2 so $y= \sqrt{4- w^2/4}$ and the height of the rectangle is $h= 2y= 2\sqrt{4- w^2/4}= \sqrt{16- w^2}$.

Since "The strenghth of a wooden beam with a rectangular cross section and fixed length is proportional to the product of its width and the square of its height where the proportionailty constant is 2 psi", strength= $2h^2w= 2(16- w^2)w= 32w- 2w^3$.

Differentiate that with respect to w to find the value of w that maximizes that.