Question
The safe load, L. of a wooden beam supported at both ends varies jointly as the width, w, the square of the depth, d, and inversely as the length, I. A wooden beam 7 in. wide. 4 in. deep, and 14 ft long holds up 9036 Ib. What load would a beam 6 in. wide, 8 in. deep and 11 ft long of the same material support? (Round off your answer to the nearest pound.)
Answers
GPT 3.5
We can set up a proportion to solve this problem.
Let L1 be the safe load of the first beam (7 in. wide, 4 in. deep, and 14 ft long) and L2 be the safe load of the second beam (6 in. wide, 8 in. deep, and 11 ft long).
According to the problem, L1 varies jointly as w, d^2, and inversely as I. So we can write the equation:
L1 = k * w * d^2 / I,
where k is the constant of variation. We can solve for k by plugging in the values given: L1 = 9036 lb, w = 7 in, d = 4 in, and I = 14 ft.
9036 = k * 7 * 4^2 / 14,
k = 9036 * 14 / (7 * 16),
k = 9036 * 2 / 16,
k = 1129.5.
Now we can use this value of k to find L2. Plugging in the values for the second beam: L2 = k * 6 * 8^2 / 11.
L2 = 1129.5 * 6 * 8^2 / 11,
L2 = 1129.5 * 6 * 64 / 11,
L2 = 54,629.45 / 11,
L2 ≈ 4975.4 lb.
Therefore, the load that the second beam can support is approximately 4975 lb when rounded to the nearest pound.
Let L1 be the safe load of the first beam (7 in. wide, 4 in. deep, and 14 ft long) and L2 be the safe load of the second beam (6 in. wide, 8 in. deep, and 11 ft long).
According to the problem, L1 varies jointly as w, d^2, and inversely as I. So we can write the equation:
L1 = k * w * d^2 / I,
where k is the constant of variation. We can solve for k by plugging in the values given: L1 = 9036 lb, w = 7 in, d = 4 in, and I = 14 ft.
9036 = k * 7 * 4^2 / 14,
k = 9036 * 14 / (7 * 16),
k = 9036 * 2 / 16,
k = 1129.5.
Now we can use this value of k to find L2. Plugging in the values for the second beam: L2 = k * 6 * 8^2 / 11.
L2 = 1129.5 * 6 * 8^2 / 11,
L2 = 1129.5 * 6 * 64 / 11,
L2 = 54,629.45 / 11,
L2 ≈ 4975.4 lb.
Therefore, the load that the second beam can support is approximately 4975 lb when rounded to the nearest pound.
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