Use a variation model to solve for the unknown value. Round your answer to the nearest whole number.
The strength of a wooden beam varies jointly as the width of the beam and the square of the thickness of the beam, and inversely as the length of the beam. A beam that is 135 in. long, 5 in. wide, and 3 in. thick can support a load of 405 lb.
Find the maximum load that can be safely supported by a board that is 10 in. wide, 270 in. long, and 6 in. thick.
First, we set up the equation to represent the variation model:
Strength = k * (width) * (thickness)^2 / length
We are given that for the original beam, the strength is 405 lb, the width is 5 in, the thickness is 3 in, and the length is 135 in. Plugging in these values, we can solve for k:
405 = k * 5 * 3^2 / 135
405 = k * 45 / 135
k = 405 * 135 / 45
k = 1215
Now that we have found k, we can find the maximum load that can be safely supported by the new beam with width 10 in, thickness 6 in, and length 270 in:
Strength = 1215 * 10 * 6^2 / 270
Strength = 1215 * 10 * 36 / 270
Strength = 43740 / 270
Strength ≈ 162 lb
Therefore, the maximum load that can be safely supported by the new beam is approximately 162 lb.