Question

A rectangular beam is cut from a cylindrical log of radius 15 cm. The strength of a beam of width w and height h is proportional to
wh^2.
Find the width and height of the beam of maximum strength. (Round your answers to two decimal places.)
h= ?
w = ?

Hello! I am struggling with this problem.
This is what I have done:

r = 15cm
h^2+w^2=30^2
so h = -w+30

and A=hw so A= w(-w+30)
so A = -w^2+30w
A'=-2w+30
-2w+30=0
so w=15

15=15h^2
so h = + or - 1

I didn't know where to go from there. Please help!

Answer 1

<i>h^2+w^2=30^2
so h = -w+30</i>
what? That's not correct.


Now, the strength s is

s = wh^2 = w(900-w^2) = 900w-w^3
so, to find the maximum strength, we need

ds/dw = 900-3w^2 = 0
This is at w^2 = 300
That means h^2 = 600

Answer 2

Thank you so much for all your help! I will make sure I study really hard since it doesn't seem like I know much about this topic.