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 216 in. long, 8 in. wide, and 3 in. thick can support a load of 428 1b.
Find the maximum load that can be safely supported by a board that is 16 in. wide, 432 in. long, and 6 in. thick.
First, write the equation based on the given information:
S = k(wt^2/l)
where S is the strength of the beam, k is the constant of variation, w is the width, t is the thickness, and l is the length.
Next, use the given values to find the value of k:
428 = k(8*3^2/216)
428 = k(8*9/216)
428 = k(72/216)
428 = k(1/3)
k = 1284
Now, use the value of k to find the maximum load that can be safely supported by the new beam:
S = 1284(16*6^2/432)
S = 1284(16*36/432)
S = 1284(576/432)
S = 1284(1.3333...)
S = 1712
Therefore, the maximum load that can be safely supported by a board that is 16 in. wide, 432 in. long, and 6 in. thick is 1712 lb.