Answer
$\text {(a) Average temperature is 22.5 degrees}$
$\text {(b) At the middle of the bar. See explanation.}$
Work Step by Step
$\text {(a) T$_1$ = 15, T$_2$ = 30 and the length is 10m.}$
$\text {Therefore, we can construct a linear function:}$
$\text {When L = 0 $\Rrightarrow$ T = 15 and when L = 10 $\Rrightarrow$ T = 30}$
$\text {Thus, the linear function is}$
\begin{align}
f(x) = 1.5x +15
\end{align}
$\text {The average value of the function is}$
\begin{align}
f_{ave} = \frac{1}{10} \int_0^{10}&(1.5x+15) \ dx = \left[ 0.75x^2+15x \right]_0^{10} = \\
& = \frac{1}{10} \times (75+150) = 22.5
\end{align}
$\text {The average temperature of the metal bar is 22.5 degrees.}$
$\text {(b)}$
$\text {Because it is an average temperature, hence, it should be between}$
$\text {the lowest and highest temperatures of the bar.}$
\begin{align}
22.5 = 1.5x+15 \Rrightarrow x = 5m
\end{align}
$\text {Meaning that the average temperature is at the middle of the bar.}$
