How Far Can I See?

How Far Can I See?


When you hear the phrase "As far as the eye can see," do you ever think about how far that really is? Can you tell how far away the horizon is from where you are standing? You may have noticed that the horizon appears farther away if you stand on a ladder or when you look out a 10th floor window.

Collecting and Modeling Data

Collecting and Modeling Data


Let's compare our height above sea level to the distance we can see to the horizon. Examine the interactive tool. As you drag the point representing your height above sea level, what do you notice about the distance you can see to the horizon?

Distance to Horizon

Create a table like the one shown below, and record at least 5 specific points. Be sure to choose points that cover a wide range of values for the height above sea level.

Height above Sea Level Distance to Horizon
   
   
   
   
   

Create a Table

Examine the data in the table; what type of function do you predict would best model this data?

Consider the point (0, 0). Does it make sense in this situation to include this point?

Graphing the points on a coordinate plane may help us determine the type of function that would best model this data. Enter your data points into the Multi-Function Data Flyer below.

Multi-Function Data Flyer

Click on Plot Data to graph the points you enter. (Note: You may have to adjust your window or choose the auto-scale feature.)

What type of function could be used to model the data?

In the next section, we will determine the function that models your data.

 

Determining the Function to Model the Data

Determining the Function to Model the Data


It appears that the data could be modeled by a square root function. In the Multi-Function Data Flyer, enter the function f (x) = 1x ^ 0.5 to represent a square root function. Note: It is necessary to enter the 1 in order to be able to adjust that value using the sliders.

Multi-Function Data Flyer

Consider the following questions:

  • How does the graph compare to the data points?
  • What parameters do you think might need to change to better match the data?

Click the "Set Function" button. Use the slider to adjust the values for the coefficient of x ^ 0.5 until you identify the function that best models the data. (Note: You may have to adjust the slider by choosing "slider limits." Recommended range of –1 to 2 and a step of 0.1)

Consider the following questions:

  • How well did your function match the data?
  • Can you estimate the distance to the horizon from the top of Mount Everest if the height of Mount Everest is 29,029 feet above sea level?
  • Can you estimate the height above sea level if the horizon is 110 miles away?