Suppose that the maximum weight that a certain type of rectangular beam can support varies inversely as its length and jointly as its width and the square of its height. Suppose also that a beam 6 inches wide, 2 inches high, and 12 feet long can support a maximum of 14 tons. What is the maximum weight that could be supported by a beam that is 4 inches wide, 3 inches high, and 14
feet long?
The strength of a beam is directly proportional to its width and the square of its depth but inversely proportional to its length. A beam that is
55
inches wide,
88
inches deep, and
1010
feet long can support a weight of
17601760
pounds. If the same type of beam that is
77
inches wide and
44
inches deep can support
440440
pounds, how long is the beam?
52 inches is what percent of 20 feet?
To solve this problem, we need to use the given information and apply the inverse variation and joint variation relationships.
Let's denote the maximum weight that the beam can support as W (in tons), the width as w (in inches), the height as h (in inches), and the length as L (in feet).
According to the problem, we know that the maximum weight varies inversely as the length and jointly as the width and the square of the height. This can be written as:
W ∝ 1/L
W ∝ w * h^2
Since W varies inversely as L, we can write:
W = k1/L where k1 is a constant.
Since W varies jointly as w and h^2, we can write:
W = k2 * w * h^2 where k2 is another constant.
Now, let's use the given information to solve for the constants.
We are given:
W = 14 tons
w = 6 inches
h = 2 inches
L = 12 feet
Substituting these values into the equation, we get:
14 = k1/12 ----(1) (we converted feet to inches for consistency)
Also,
14 = k2 * 6 * (2^2)
14 = k2 * 6 * 4
14 = 24k2 ----(2)
Now, let's solve equations (1) and (2) simultaneously to find the values of k1 and k2.
From equation (1), we can solve for k1:
k1 = 14 * 12
k1 = 168
Substituting the value of k2 into equation (2), we get:
14 = 24k2
14 = 24 * (1/2)
14 = 12
Now that we have determined the values of k1 and k2, we can use these values to find the maximum weight that could be supported by the new beam.
The new beam has the following dimensions:
w = 4 inches
h = 3 inches
L = 14 feet
Using the equation W = k2 * w * h^2 with the values of k2, w, and h, we can calculate the maximum weight:
W = 12 * 4 * (3^2)
W = 12 * 4 * 9
W = 432 tons
Therefore, the maximum weight that could be supported by a beam that is 4 inches wide, 3 inches high, and 14 feet long is 432 tons.