SOLUTION
Given the question in the image, the following are the solution steps to answer the question
STEP 1: Write the daily routes for the first 8 days
![\begin{gathered} 105\operatorname{km},82\operatorname{km},68\operatorname{km},94\operatorname{km},54\operatorname{km},102\operatorname{km},88\operatorname{km},73\operatorname{km} \\ \text{ number of days=8} \\ n=8 \end{gathered}]()
STEP 2: Compute the arithmetic mean for the first 8 days
![\begin{gathered} \operatorname{mean}(\bar{x})=\frac{\sum^{}_{}\text{routes for first 8 days}}{nu\text{mber of days}} \\ \operatorname{mean}(\bar{x})=(105+82+68+94+54+102+88+73)/(8)=(666)/(8)=83.25\operatorname{km} \end{gathered}]()
The mean of the distances of these 8 days is 83.25km
STEP 3: Compute the median of the distances of these 8 days
To find the Median, place the numbers in ascending order and find the middle number.
![\begin{gathered} 105\operatorname{km},82\operatorname{km},68\operatorname{km},94\operatorname{km},54\operatorname{km},102\operatorname{km},88\operatorname{km},73\operatorname{km} \\ By\text{ arranging these distances in ascending order, we have;} \\ 54km,68\operatorname{km},73\operatorname{km},82\operatorname{km},88\operatorname{km},94\operatorname{km},102\operatorname{km},105\operatorname{km} \\ We\text{ now look for the middle distance} \\ \text{Since the data are even, we pick the two middle distances and divide by 2} \\ \Rightarrow82\operatorname{km}\text{ and 88km are the middle distances} \\ \operatorname{median}=(82+88)/(2)=(170)/(2)=85\operatorname{km} \end{gathered}]()
The median of the distances of these 8 days is 85km
STEP 4: Calculate the average distance needed in order to reach her goal
![\begin{gathered} \text{Her goal}\Rightarrow\text{average of 80km per day} \\ \text{First 8 days}\Rightarrow\text{average of 83.25 per day(From step 2)} \\ Total\text{ vacation days=14} \\ \text{average goal=}\frac{average\text{ of first 8 days+average of last 6 days}}{2} \\ By\text{ substitution,} \\ 80=(83.25+x)/(2) \\ By\text{ cross multiplication,} \\ 80*2=83.25+x \\ 160=83.25+x \\ \text{Subtarct 83.25 from both sides} \\ 160-83.25=83.25+x-83.25 \\ 76.75=x \\ x=76.75\operatorname{km} \end{gathered}]()
Therefore, the average distance needed in the remaining days in order to reach her goal is 76.75km