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!

h^2+w^2=30^2

so h = -w+30
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

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.

To find the width and height of the beam that will maximize its strength, we can use the derivative to find the critical points. Let's continue from where you left off:

To find the critical points, we need to solve A' = -2w + 30 = 0 for w.

-2w + 30 = 0
2w = 30
w = 15

So, we have found that the critical width value is 15 cm.

Now, to find the height, we can substitute w = 15 into the equation h^2 + w^2 = 30^2:

h^2 + 15^2 = 30^2
h^2 + 225 = 900
h^2 = 900 - 225
h^2 = 675

Taking the square root of both sides:

h = √675 ≈ 26.04 or h = -√675

Since the height cannot be negative in this context, we can discard the negative value. Therefore, the height is approximately h = 26.04 cm.

So, the width of the beam of maximum strength is w ≈ 15 cm, and the height is h ≈ 26.04 cm.

To find the width and height of the beam of maximum strength, we need to maximize the function A = wh^2, where w is the width and h is the height.

You correctly found the expression for the area of the beam: A = -w^2 + 30w. However, there seems to be a sign mistake in your derivative calculation.

Let's go through the steps again:

1. Express the area as a function of one variable:
A = -w^2 + 30w

2. Take the derivative of the area function with respect to w:
A' = -2w + 30

3. Set A' equal to zero and solve for w:
-2w + 30 = 0
2w = 30
w = 15

So the width of the beam with maximum strength is 15 cm.

Now, to find the corresponding height, substitute this value of w back into the equation you derived earlier relating w and h: h^2 + w^2 = 30^2

15^2 + h^2 = 30^2
225 + h^2 = 900
h^2 = 900 - 225
h^2 = 675
h = √675
h ≈ 25.98 cm (rounded to two decimal places)

Therefore, the height of the beam with maximum strength is approximately 25.98 cm.

To recap:
Width (w) = 15 cm
Height (h) ≈ 25.98 cm (rounded to two decimal places)