The maximum load of a horizontal beam that is supported at both ends varies jointly as the width and the square of the height and inversely as the length between the supports. A beam 6 m long, 0.1 m wide, and 0.06 m high supports a load of 360 kg. When the maximum load supported by a beam 16 m long and 0.2 m wide is 480 kg, what is its height?

To find the height of the beam, we can use a joint variation equation and a proportion. Using the given values, we first find the constant of variation, k. Then, we can use this k value to solve for the height in the second scenario when the maximum load is given. The height of the beam is approximately 0.122 m.

Explanation:

Let's denote the maximum load as L, the width of the beam as w, the height as h, and the length between the supports as l. The given information tells us that when w = 0.1 m, h = 0.06 m, and l = 6 m, L = 360 kg. We can use this information to create a proportion:

L / (w * h^2) = k / l

where k is a constant of variation. Substituting the given values, we have:

360 / (0.1 * 0.06^2) = k / 6

Simplifying, we find k = 777600.

Now we can use this k value to find the height, h, of the beam when

L = 480 kg,

w = 0.2 m, and

l = 16 m:

480 / (0.2 * h^2) = 777600 / 16

Simplifying and solving for h, we find:

h^2 = 0.015

h = √0.015

h ≈ 0.122 m

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