The simply supported joist is used in the construction of a floor for a building. In order to keep the floor low with respect to the sill beams C and D, the ends of the joists are notched as shown. If the allowable shear stress for the wood is psi and the allowable bending stress is psi, determine the smallest height h so that the beam will support a load of . Also, will the entire joist safely support the load? Neglect the stress concentration at the notch.

Definitionb 1.1. Given two integers a and d with d non-zero, we say that d divides a (written d | a) if there is an integer c with a = cd. If no such integer exists, so d does not divide a, we write d - a. If d divides a, we say that d is a divisor of a. Proposition 1.2.1: Assume that a, b, and c are integers. If a | b and b | c, then a | c. Proposition 1.3. Assume that a, b, d, x, and y are integers. If d | a and d | b then d | ax + by. Corollary 1.4. Assume that a, b, and d are integers. If d | a and d | b, then d | a + b and d | a − b. Prime: A prime number is an integer p ≥ 2 whose only divisors are 1 and p. A composite number is an integer n ≥ 2 that is not prime. Ex: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. The Division Algorithm: Let a and b be integers with b > 0. Then there exist