49. The function \( f(x) = \begin{cases} 1 + \frac{3}{4}x & \text{for } 0 \leq x < 4 \\\\ 5 - \frac{1}{4}(x - 6)^2 & \text{for } 4 \leq x < 8 \\\\ 4 & \text{for } 8 \leq x \leq 10 \end{cases} \)
A mountain hike consists of a steady incline followed by a curved hill and then a flat valley. The mountain hike is modeled by the piecewise-defined function \( f \) above, and the graph of \( f \) is shown in the figure above. Which of the following expressions gives the total length of the hike from \( x = 0 \) to \( x = 10 \)?
Answer is: option4
Solution:
\( 7 + \int_4^8 \sqrt{1 + \frac{1}{4}(x - 6)^2} \, dx \)
