A rectangular beam is cut from a cylindrical log of radius 30 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.
Is the width 45 and the height 20.
Hey, the height is surely more than the width for strength
w = 1/2 width
h = 1/2 height
15 = sqrt(w^2+h^2)
w^2+h^2 = 225
so
h^2 = 225 - w^2
then
strength = s = k w h^2 = k w(225-w^2)
find where ds/dw = 0
s = k (225 w - w^3)
ds/dw = k (225 - 3 w^2) =0
w^2 = 225/3 = 75
then h^2 = 225 -75 = 150
so
w = 5 sqt3 and width = 10 sqrt 3
h = 5 sqrt 6 and length = 10 sqrt 6
by the way
h/w = sqrt 2 = 1.41
To find the width and height of the beam of maximum strength, we need to maximize the strength function wh^2.
Let's denote the width as w and the height as h.
Given that the beam is cut from a cylindrical log of radius 30 cm, we can consider the cross-section of the log as a rectangle. The width of the rectangle will be equal to the diameter of the cylinder, which is twice the radius, or 2 * 30 cm = 60 cm.
So, we have w = 60 cm.
Now, the strength function is proportional to wh^2, which can be written as S = kwh^2, where k is the constant of proportionality.
Since we want to find the maximum strength, we need to maximize S. We can do this by finding the values of w and h that maximize S.
To do this, we must differentiate S with respect to both w and h, and set both partial derivatives equal to zero.
∂S/∂w = kh^2 = 0 (1)
∂S/∂h = 2kwh = 0 (2)
From equation (2), we see that either k = 0, w = 0, or h = 0. However, if either w or h is zero, the strength would also be zero. So we can ignore these cases.
Plugging w = 60 cm into equation (1), we have:
kh^2 = 0
60h^2 = 0
Since k ≠ 0, we get h^2 = 0, which means h = 0. But h cannot be zero.
So, the dimensions w = 60 cm and h = 0 cm do not satisfy the condition of maximizing the strength.
Hence, the values of width and height you suggested, w = 45 and h = 20, are incorrect.
To find the values of width and height that maximize the strength, we need further information or a different approach.
To find the width and height of the beam with maximum strength, we need to maximize the function wh^2 with respect to w and h.
Since the beam is cut from a cylindrical log, the width (w) of the beam will be equal to the diameter of the log, which is twice the radius. Hence, w = 2 * 30 cm = 60 cm.
Now, we need to find the height (h) of the beam. Let's assume the height of the beam is x.
Since the beam is rectangular, the volume (V) of the beam can be calculated as length (l) multiplied by width (w) multiplied by height (h):
V = lwh
But since the log is cylindrical, the volume of the beam will be equal to the volume of the log. Hence,
V = π(30^2) * h
Now, we need to relate the volume of the beam (V) to its width (w) and height (h) using the given strength formula: wh^2. We can set up the following equation:
wh^2 = V
Substituting the expressions for V and w, we get:
(60 cm)(x^2) = π(30^2) * h
To find the value of x that maximizes the strength, we need to differentiate the above equation with respect to x and equate it to 0. Differentiating the equation, we get:
120x = π(30^2)
Simplifying this equation,
x = π(30^2) / 120
Evaluating this expression, we find:
x ≈ 23.56 cm
Thus, the width of the beam is w = 60 cm and the height of the beam is h ≈ 23.56 cm.
Therefore, the width mentioned (45 cm) and the height (20 cm) are not correct.