The total distance d remaining on a bike ride after t hours can be modeled with a linear function.

Question

The person has 18 miles left on the ride after 1.5 hours, and 13 miles left after 3.5 hours. 

A) Give a linear function which represents d as a function of t. Make sure to use good function notation.

B) How much further must they ride after 2 hours?

C) When will they have 10 miles left?

D) How much total distance will they ride?

E) How long will it take them to complete the ride?

Answer 1

distance remaining as a function of time

d(t) = mt + b

where m = slope, b = y-intercept, t = time elapsed

 

We have two points:  (1.5, 18), (3.5, 13)

Find the slope:  m = (13-18)/(3.5-1.5) = -5/2 = -2.5

This means that for every hour that passes we have 2.5 less miles to travel.

 

d(t) = -2.5t + b

Plugging in the point (1.5, 18) for (t, d(t)) find b

18 = -2.5(1.5) + b

18 = -3.75 + b

b = 21.75

NOTE:  THIS MEANS AT TIME T = 0 THEY HAD 21.75 MILES TO GO

 

Your linear function is:  d(t) = -2.5t + 21.75

 

B) How much further must they ride after 2 hours?

d(2) = -2.5(2) + 21.75

       = -5 + 21.75

       = 16.75 miles left after 2 hours

 

C)  When will they have 10 miles left?

     10 = -2.5t + 21.75

     Solve for t

 

D) How much total distance will they ride?

     This has already been answered in part A by finding the y-intercept...21.75 miles

 

E)  How long will it take them to complete the ride?

     This equates to:  Find the value of t when d(t) = 0

     0 = -2.5t + 21.75

     Solve for t