The strength of a rectangular beam is proportional to width*depth^2. What are the dimensions of the strongest rectangular beam that can be cut out of a 12 inch diameter log?
let the width be w, and the depth be d
then
S = kwd^2 where k is a constant
but d^2 + w^2 = 144
d^2 = 144-w^2
S = kw(144-w^2)
= 144kw - kw^3
d(S)/dw = 144k - 3kw^2
= 0 for max of S
3kw^2 = 144k
w^2 = 48
w = √48
then d = .....
I will let you finish up.
To find the dimensions of the strongest rectangular beam, we need to maximize the strength, which is proportional to width multiplied by depth^2.
Let's assume that the rectangular beam has a width of W and a depth of D. Since it will be cut out of a 12-inch diameter log, the width and depth must fit within the log's diameter.
We can express the width and depth in terms of the radius, which is half of the diameter:
Radius (r) = Diameter / 2 = 12 inches / 2 = 6 inches
We need to find the dimensions of the rectangle with the maximum strength. Therefore, we need to maximize the function:
Strength = Width * Depth^2
Since the strength is proportional to width * depth^2, we have:
Strength = k * Width * Depth^2
where k is a constant that we can ignore for now since it won't affect the relative maximization.
Now, let's express the width and depth in terms of the radius:
Width = W
Depth = D
We know that the rectangular beam must fit within the log, meaning its width and depth must be smaller than or equal to the diameter. Therefore:
W ≤ 2r = 2 * 6 inches = 12 inches
D ≤ 2r = 2 * 6 inches = 12 inches
Now we need to find the maximum values of W and D such that W * D^2 is maximized.
Since W and D are both less than or equal to 12 inches, we can conclude that the maximum values for W and D are 12 inches. Therefore, the maximum dimensions of the strongest rectangular beam that can be cut out of a 12 inch diameter log are:
Width = 12 inches
Depth = 12 inches
So, a rectangular beam with dimensions 12 inches by 12 inches will provide the greatest strength when cut from a 12-inch diameter log.
To find the dimensions of the strongest rectangular beam, we need to maximize the strength function, which is proportional to width times depth squared.
We are given a 12-inch diameter log, so the maximum width we can have is 12 inches.
Let's assume the width is w inches and the depth is d inches.
The strength function can be written as S = w * d^2.
Since we are looking for the strongest rectangular beam, we need to maximize this function.
To find the maximum, we can take the derivative of the strength function with respect to d and set it equal to zero.
dS/dw = d(w * d^2)/dw = d^2 + 2wd * dd/dw
Setting this equal to zero, we get d^2 + 2wd * dd/dw = 0
Simplifying, we have d(dd/dw) = -d^2/2w
Integrating both sides, we get dd/dw = -d^2/2w + C
Where C is the constant of integration.
We can rewrite this as dd/dw = -(d^2)/(2w) + C
To solve for C, we can use the fact that the width can be at most 12 inches, so when w = 12, d must be half of that, i.e., d = 6 inches.
Substituting these values into the equation, we get dd/dw = -36/(2*12) + C
Simplifying, we have dd/dw = -3 + C
Since this is a constant, let's set C = 3.
Thus, dd/dw = -3 + 3 = 0
This means that the rate of change of the rate of change of S with respect to w is zero, indicating a maximum point.
So, for maximum strength, the rate of change of the strength function with respect to width should be zero.
Now, let's solve for w and d.
From the equation dd/dw = -3, we have d = -3w + K, where K is another constant.
Using the fact that d = 6 inches when w = 12 inches, we can substitute these values into the equation and solve for K.
6 = -3(12) + K
6 = -36 + K
K = 42
Therefore, d = -3w + 42
To maximize the strength, we need to find the values of w and d that satisfy the equation dd/dw = 0.
Substituting d = -3w + 42 into dd/dw = 0, we get (-3w + 42)d/dw = 0
Simplifying, we have -3w^2 + 42w = 0
Factoring out w, we get w(-3w + 42) = 0
Setting each factor equal to zero, we have w = 0 and -3w + 42 = 0
The width cannot be zero, so we solve the second equation for w:
-3w + 42 = 0
-3w = -42
w = -42/-3
w = 14
So, the width of the strongest rectangular beam is 14 inches.
To find the depth, we can substitute this value of w into d = -3w + 42:
d = -3(14) + 42
d = -42 + 42
d = 0
Thus, the dimensions of the strongest rectangular beam that can be cut out of a 12-inch diameter log are a width of 14 inches and a depth of 0 inches.