1. How many inches per minute does London's elevation change between 4 minutes and 8 minutes.
The question actually asks for the slope of the line that stands for the points [tex](4,3) \ and \ (8,6)[/tex] why? because the questions tells us that London's elevation changes between 4 minutes and 8 minutes here. Hence, to find the slope of this line we have to use the following formula:
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ \\ But: \\ \\ P(x_{1},y_{1})=P(4,3) \\ P(x_{2},y_{2})=P(8,6)[/tex]
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ \\ But: \\ \\ P(x_{1},y_{1})=P(4,3) \\ P(x_{2},y_{2})=P(8,6) \\ \\ So: \\ \\ m=\frac{6-3}{8-4}=0.75in/min[/tex]
So London's elevation changes 0.75 inches per minute
2. During which time period does London's elevation change the fastest?
The greater the absolute value of the slope of the line the faster London's elevation changes. Since this is a Piecewise function, we must analyze each period.
→ Between 0 minutes and 4 minutes the function is constant, so there is no any change here.
→ Between 10 minutes and 14 minutes the function is constant, so there is no any change here.
→ Between 18 minutes and 22 minutes the function is constant, so there is no any change here.
So the solution is not in these parts of the function.
→ Between 4 minutes and 10 minutes the function has a positive slope, so there is change here.
In the previous item we calculated the slope between 4 and 8 minutes that is the same slope between 4 and 8 minutes and equals 0.75.
→ Between 14 minutes and 18 minutes the function has a positive slope, so there is change here.
Let's take two points here, say, [tex](16,5) \ and \ (18,3)[/tex]
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ \\ But: \\ \\ P(x_{1},y_{1})=P(16,5) \\ P(x_{2},y_{2})=P(18,3) \\ \\ So: \\ \\ m=\frac{3-5}{18-16}=-1 in/min[/tex]
As you can see, the absolute value here is 1 that is greater than 0.75.
In conclusion, London's elevation changes the fastest between 14 and 18 minutes
