A railroad bridge is built over a V-shaped canyon. If the supports are 8 feet long at each end, the center support is 560 feet long, and the supports increase in length by 4 feet with each support (from 8 feet to 560 feet),
What is the total length of all the supports?
A) 78,384 ft
B) 78,392 ft
C) 78,396 ft
D) 78,388 ft

Question

Quizplus · Accepted Answer

The sequence of the lengths of the supports forms an arithmetic sequence, starting from 8 feet, increasing by 4 feet each time, up to 560 feet. The total number of supports can be found by using the formula for the nth term of an arithmetic sequence: $a_n = a_1 + (n - 1)d$, where $a_n$ is the nth term, $a_1$ is the first term, $d$ is the common difference, and $n$ is the number of terms. Solving for $n$, we get $560 = 8 + (n - 1)4$, which simplifies to $552 = (n - 1)4$, and further to $138 = n - 1$, giving $n = 139$. This means there are 139 supports on one side of the canyon, and since the pattern is symmetric, there are a total of $139 \times 2 = 278$ supports, not forgetting to add one more for the center support, making 279 supports in total.The sum of an arithmetic series is given by $S_n = \frac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum of the first $n$ terms. For one side, the sum is $S_{139} = \frac{139}{2}(8 + 560)$. For both sides together, the total length is $2 \times S_{139} + 560$ (adding the center support).Calculating $S_{139}$ gives us $S_{139} = \frac{139}{2}(568) = 39456$, and for both sides, $2 \times 39456 = 78912$. Adding the center support of 560 feet gives a total of $78912 + 560 = 78,472$. However, this calculation seems to have an error in the final addition step. Correctly calculating the total length involves understanding that the sequence's sum calculation should directly lead to the total length of all supports, including the center, without needing to add the center support's length again after doubling the sum of one side.The correct approach is to find the total number of supports (279), calculate the sum of the lengths of all supports directly, and ensure the arithmetic series formula is applied correctly. Given the discrepancy in the final step of the explanation, let's correct the calculation:The correct sum for all supports (including both sides and the center) is calculated by considering the entire sequence as one. Since there are 279 supports in total, we use the formula for the sum of an arithmetic series: $S = \frac{n}{2}(a_1 + a_n)$, where $n = 279$, $a_1 = 8$, and $a_n = 560$.Thus, the correct sum is $S = \frac{279}{2}(8 + 560)$, which simplifies to $S = 139.5 \times 568$. This calculation aims to find the total length of all supports, but the correct answer must match one of the provided options, which was initially calculated based on the progression from 8 feet to 560 feet with each support increasing by 4 feet.Given the mistake in the explanation regarding the final addition and the correct method to calculate the total length, the focus should be on correctly applying the arithmetic series sum formula to the entire set of supports, considering the progression and the total number of supports. The initial setup suggests an arithmetic progression, but the final step should accurately reflect the sum of this progression without incorrectly adding the center support post-calculation. The correct answer should directly follow from the sum of the arithmetic series formula applied to all supports, considering the progression from 8 feet up to 560 feet with the specified increment.

A Railroad Bridge Is Built Over a V-Shaped Canyon