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?
w = k(1/l)(wh^2)
for the given:
14 = k(1/12)(6(2^2))
14 = k(2)
k = 7
so w = 7(wh^2/l)
= (7/14)(4(9))
= 18 tons
Thank you!
The maximum weight that a rectangular beam can support varies jointly as its width and the square of its height and inversely as is length. Of a beam 1/2 foot wide,1/3 foot high, and 10 feet long can support 12 tons, find how much a similar beam can support if the beam is 2/3 foot wide, 1/2 foot high, and 16 feet long.
To solve this problem, we need to use the inverse variation formula and the joint variation formula.
Let's assign variables to the different quantities mentioned in the problem:
Let x be the maximum weight in tons.
Let l be the length of the beam in feet.
Let w be the width of the beam in inches.
Let h be the height of the beam in inches.
According to the problem, the maximum weight is inversely proportional to the length and directly proportional to the width and the square of the height.
This gives us the following equation:
x = k * (1/l) * (w * h^2)
We are given the values for a beam with specific dimensions (6 inches wide, 2 inches high, and 12 feet long) that can support a maximum weight of 14 tons.
Using this information, we can solve for the constant k:
14 = k * (1/12) * (6 * 2^2)
14 = k * (1/12) * (6 * 4)
14 = k * (1/12) * 24
14 = 2k
k = 7
Now that we know the value of k, we can use it to find the maximum weight that could be supported by a beam with different dimensions (4 inches wide, 3 inches high, and 14 feet long).
Let's substitute the values into the equation:
x = 7 * (1/14) * (4 * 3^2)
x = 7 * (1/14) * (4 * 9)
x = 7 * (1/14) * 36
x = 7 * 2
x = 14
The maximum weight that could be supported by a beam that is 4 inches wide, 3 inches high, and 14 feet long is 14 tons.