Sandia Aerial Tram

Task

Albuquerque boasts one of the longest aerial trams in the world. The tram transports people up to Sandia Peak. The table shows the elevation of the tram at various times during a particular ride.

1. Write an equation for a function (linear, quadratic, or exponential) that models the relationship between the elevation of the tram and the number of minutes into the ride. Justify your choice.
2. What was the elevation of the tram at the beginning of the ride?
3. If the ride took 15 minutes, what was the elevation of the tram at the end of the ride?

IM Commentary

The time values are irregularly spaced to prompt a more detailed analysis of the rate of change of the elevation.

The task provides an opportunity for students to engage in Mathematical Practice Standard 4: Model with mathematics.

Solution

1. The average rate of change in height with respect to time between each column in the table is the same: \begin{equation} \frac{7834-7069}{5-2}=\frac{8854-7834}{9-5}=\frac{10129-8854}{14-9}=255.\end{equation} Therefore we choose a linear function to model the relationship. If $y$ represents the elevation of the tram in feet and $x$ represents the number of minutes into the trip, then $y-7069=255(x-2)$ for each pair $(x,y)$ in the table, so the linear function given by $y = 7069 +255(x-2)$ works.
2. When $x = 0$ $$ y = 7069 + 255(0-2) = 6559.$$ So the elevation of the tram at the beginning of the ride is 6559 feet.
3. When $x=15$ $$ y=7069+255(15-2)=10,384.$$ So the elevation of the tram at the end of the ride is 10,384 feet.