You all know that going up a hill slows down your pace, but by how much?

We will try and provide a mathematical model to describe this situation.

Problem 1

What factors will affect you average walking time up a hill?

In this analysis, we will assume that the walking time depends on four factors:

• horizontal distance travelled (map distance)        d     miles
• vertical distance travelled (height of hill)              h     feet
• speed for horizontal walking                               x     mph
• speed for vertical climbing                                  y     feet per hour

The model will be assumed to be of the form

     where T is the time taken.

Problem 2   Use the experimental data given below to find the approximate values for x and y.

Map distance	Height climbed	Walking time
d (miles)	h (feet)	T (hours)
12	1500	5½
15	2000	7

Problem 3   Use the model to determine how long it might take to climb up Mount Snowdon, 3560 feet high, if the horizontal distance from your starting point is 3 miles.

Problem 4   If it takes 8 hours to climb a hill and you map distance is 4 miles, estimate the height of the hill.

EXTENSIONS

1.   Apply this test to a local situation, using your own experimental data, to see if this model works in practice.

2.   Can you produce a model for the time it takes to come down a hill?