An engineer wants to build a staircase with 9 bar whose lengths decrease uniformly from the base to the top. The 2nd bar measures 22.75 inches and the penultimate bar measures 15.25 inches. Determine the sum of the lengths of all the bars. A) 156 inches total B 164 inches overall 171 inches total 182 inches total

To determine the sum of the lengths of all the bars in the staircase, you can use the concept of an arithmetic progression since the lengths decrease uniformly from the base to the top. In an arithmetic progression, each term is obtained by adding a common difference to the previous term. Let's denote the length of the first bar as 'a' and the common difference between consecutive bars as 'd'. According to the information given: - The 2nd bar measures 22.75 inches, so the 2nd term is 22.75: a + d = 22.75 - The penultimate (8th) bar measures 15.25 inches, so the 8th term is 15.25: a + 7d = 15.25 Now, we have a system of two equations with two unknowns: 1. a + d = 22.75 2. a + 7d = 15.25 We can solve this system of equations simultaneously to find the values of 'a' and 'd'. First, subtract the first equation from the second equation to eliminate 'a': (a + 7d) - (a + d) = 15.25 - 22.75 6d = -7.5 Now, divide both sides by 6 to solve for 'd': d = -7.5 / 6 d = -1.25 inches Now that we have found 'd', we can find 'a' using the first equation: a + (-1.25) = 22.75 a = 22.75 + 1.25 a = 24 inches So, the first bar has a length of 24 inches, and the common difference is -1.25 inches. To find the sum of the lengths of all the bars, you can use the formula for the sum of an arithmetic progression: Sum = (n/2) * [2a + (n-1)d] In this case, n is the number of bars (which is 9), 'a' is 24 inches, and 'd' is -1.25 inches. Sum = (9/2) * [2 * 24 + (9-1) * (-1.25)] Sum = (9/2) * [48 - 10] Sum = (9/2) * 38 Sum = 9 * 19 Sum = 171 inches total So, the sum of the lengths of all the bars in the staircase is 171 inches, which corresponds to option C) 171 inches total.