A straight fence is to be constructed from posts 6 inches wide and separated by lengths of chain 5 feet long. If a certain fence begins and ends with a post, which of the following could not be the length of the fence in feet? (12 inches = 1 foot)
A. 17
B. 28
C. 35
D. 39
E. 50
Answer (3)
Well, to get the possible length you start with 6 in (0.5 feet) and add 5.5 feet multiple times. You could get 6, 11.5, 17, 22.5, 28, 33.5, 39, 44.5, 50, ... Notice we missed the "35" option, which is, thus, not possible.
The correct answer is C. 35, as it is the only length that does not conform to the repeating pattern of post and chain lengths when using the formula L = 5n + 0.5(n+1). Lengths that are not a multiple of 5.5 plus an additional 0.5 feet due to the starting post are invalid options.
The question involves a fence constructed from posts and chains. Each post is 6 inches wide (which is 0.5 feet), and each length of chain is 5 feet long.
To calculate the possible lengths of the fence, one must remember that the fence begins and ends with a post. Thus, the length of the fence will consist of a series of 5-foot chains and 0.5-foot posts. If n is the number of chains, there must be n+1 posts to end with a post.
The length (L) of the fence can be expressed by the formula:
L = 5n + 0.5(n+1)
Let's try to find a combination that does not result in a whole number:
For n=1: L = 5(1) + 0.5(1+1) = 5 + 1 = 6 feet
Increasing n by 1 will always add 5.5 feet to the total length of the fence.
Starting from 6 feet, the addition of 5.5 feet increments can only yield lengths that, when divided by 5.5, result in a whole number plus an additional 0.5 feet due to the starting post. Therefore, any length that is not a multiple of 5.5 plus 0.5 cannot be a length of the fence.
Checking the options:
Option C is the only one that does not result in a whole number after the calculation, meaning a fence of 35 feet doesn't fit the given pattern of post and chain lengths.
The only length that could not be constructed using the specified posts and chain lengths is option D, 39 feet. This is because it does not align with the formula for total length derived from the given setup. All other options can be constructed with the provided posts and chains.
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