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------------------------------------------------ The strength \( S \) of a rectangular beam varies jointly as its width \( w \) and the square of its thickness \( t \). If a wood beam 5 inches wide and 6 inches thick supports 450 pounds, how much can a similar beam 5 inches wide and 4 inches thick support?

The strength S of a rectangular beam varies jointly as its width w and the square of its thickness t. This means that the strength is directly proportional to the width and the square of the thickness.
To find out how much a similar beam with a width of 5 inches and a thickness of 4 inches can support, we can set up a proportion using the given information.
Let's call the strength of the first beam S1 and the strength of the second beam S2.
We know that S1 = 450 pounds, w1 = 5 inches, t1 = 6 inches, w2 = 5 inches, and t2 = 4 inches.
Using the joint variation equation, we have:
S1 = k * w1 * t1^2
S2 = k * w2 * t2^2
Since both beams are similar, we can set up a proportion:
S1/S2 = (w1 * t1^2) / (w2 * t2^2)
Substituting the given values, we have:
450/S2 = (5 * 6^2) / (5 * 4^2)
450/S2 = (5 * 36) / (5 * 16)
450/S2 = 36/16
Cross multiplying, we get:
36 * S2 = 450 * 16
Dividing both sides by 36, we find:
S2 = (450 * 16) / 36
Simplifying, we have:
S2 = 200 pounds
Therefore, a similar beam with a width of 5 inches and a thickness of 4 inches can support 200 pounds.

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