Four rods have a length range of 3 meters. Their coefficients of linear expansion are $\alpha$, $2\alpha$, $4\alpha$, and $6\alpha$ respectively. These rods are used to separately form composite rods of lengths 7 meters and 11 meters. Calculate the average coefficient of linear expansion for each composite rod and find the value of $\alpha$ if the highest length is 6 meters. We assume the question is asking for the coefficient of expansion $\alpha$ if the maximum single rod length is 6 meters. The 4 rods have a length range of 3 meters. This implies the difference between the maximum and minimum rod lengths is 3 meters.

1. Problem Description

Four rods have a length range of 3 meters. Their coefficients of linear expansion are $\alpha$, $2\alpha$, $4\alpha$, and $6\alpha$ respectively. These rods are used to separately form composite rods of lengths 7 meters and 11 meters. Calculate the average coefficient of linear expansion for each composite rod and find the value of $\alpha$ if the highest length is 6 meters. We assume the question is asking for the coefficient of expansion $\alpha$ if the maximum single rod length is 6 meters. The 4 rods have a length range of 3 meters. This implies the difference between the maximum and minimum rod lengths is 3 meters.

2. Solution Steps

Let the lengths of the rods be $l_1$, $l_2$, $l_3$, and $l_4$ with respective coefficients of linear expansion $\alpha$, $2\alpha$, $4\alpha$, and $6\alpha$.

The lengths are proportional to the coefficients.

Let the constant of proportionality be $k$. Then $l_1=k\alpha, l_2=2k\alpha, l_3=4k\alpha, l_4=6k\alpha$.

The range of the lengths is given as 3m. The maximum length is $6k\alpha$ and the minimum length is $k\alpha$. Therefore,

$6k\alpha - k\alpha = 3$

$5k\alpha = 3$

$k\alpha = \frac{3}{5}$

The maximum individual rod length is given as 6m. Hence, $6k\alpha = 6$.

Substituting $k\alpha = \frac{3}{5}$ in the above equation, we have $6(\frac{3}{5}) = 6$. This is not correct.

Alternatively, from $6k\alpha = 6$, we get $k\alpha = 1$.

From $5k\alpha = 3$, we have $k\alpha = \frac{3}{5}$.

If $6k\alpha = 6$ is the maximum length, then the length of the rods are $k\alpha, 2k\alpha, 4k\alpha, 6k\alpha$. The lengths are then $1, 2, 4, 6$.

The range of the lengths is $6 - 1 = 5$, which is not 3 as originally stated. This means the individual rod lengths are not necessarily those associated with the coefficients of expansion.

We are given that the range of lengths of the 4 rods is 3 meters. Let the minimum length be $L$. Then the lengths of the rods are $L, L+x, L+y, L+3$, where $0 < x < y < 3$. We are given that the coefficients of linear expansion are $\alpha, 2\alpha, 4\alpha, 6\alpha$. Since there is no relationship between lengths and coefficients of expansion given, we can only say the rods are mixed to achieve rods of lengths 7 and

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Since the maximum length of any rod is 6 meters, this means $L+3 = 6$, or $L=3$. So the range is from 3 meters to 6 meters. The difference in length of the rods is still 3m.

Since the rod lengths do not depend on $\alpha$, the fact that the range of length of the rods is 3 meters and the largest single length is 6 meters allows us to state that $\alpha$ has nothing to do with length.

If we want to determine $\alpha$, we would need to solve the composite rods. The composite rods are 7 meters and 11 meters, and we do not know the percentage of rods of different $\alpha$ coefficient, so we cannot get a weighted average without this information.

$6k\alpha = 6$

$k\alpha = 1$

$5k\alpha = 3$

$5(1) = 3$, not equal. There is something missing in the question.

If we consider only the information given (4 rods with coefficients of linear expansion $\alpha, 2\alpha, 4\alpha, 6\alpha$) and the length constraint for the single largest rod is 6, then

$L_i = c_i \alpha$ where $c_i$ takes values from $(1, 2, 4, 6)$ and $6 \alpha \le 6$

$6 \alpha = 6$ (Max length is 6)

$\alpha = 1$.

$4 \alpha$, $2 \alpha$, $\alpha$ can have any values.

If $\alpha=1$ then the lengths of the rods could be 1, 2, 4, 6, but their difference needs to be

3. Since the range is 6-1=5, $\alpha \neq 1$.

The question is incorrectly written. It's not clear what it wants. If we ASSUME that largest rod is the one with $6\alpha$, we can set that to the highest length (corrected by the user). Thus, if $6\alpha=6$, $\alpha=1$.

3. Final Answer

$\alpha = 1$