Question

1. the jones family is travelling to the beach. the youngest child, samantha, determines that the distance from their home is 110 km.

a) how long will the trip take if mr. jones averages 75 km/h? 88 km/h? 105 km/h?
b) write a function that describes the time it takes to make this trip as a function of the speed of the vehicle. identify the meaning of the variables.
c) graph the function determined in part b). show asymptotes.
d) what does this graph tell you about the time it will take to travel, depending on the speed of the car?

Explanation:

Step1: Calculate time for different speeds

Use the formula \(t=\frac{d}{v}\), where \(d = 110\) km.

• For \(v = 75\) km/h:

\(t=\frac{110}{75}=\frac{22}{15}\approx1.47\) h

• For \(v = 88\) km/h:

\(t=\frac{110}{88}=\frac{5}{4} = 1.25\) h

• For \(v = 105\) km/h:

\(t=\frac{110}{105}=\frac{22}{21}\approx1.05\) h

Step2: Write the function

Let \(t\) be the time (in hours) and \(v\) be the speed (in km/h). The function is \(t(v)=\frac{110}{v}\), where \(v>0\) (speed cannot be zero or negative).

Step3: Graph analysis (brief)

The function \(t(v)=\frac{110}{v}\) is a hyperbola. The vertical asymptote is \(v = 0\) (since speed \(v\) cannot be zero) and the horizontal asymptote is \(t = 0\) (as \(v\to\infty\), \(t\to0\)).

Step4: Interpretation of the graph

As the speed \(v\) of the car increases, the time \(t\) taken to travel decreases. The relationship is inverse - the higher the speed, the less time it takes to cover the 110 - km distance.

Answer:

a) For \(v = 75\) km/h, \(t\approx1.47\) h; for \(v = 88\) km/h, \(t = 1.25\) h; for \(v = 105\) km/h, \(t\approx1.05\) h.
b) Function: \(t(v)=\frac{110}{v}\), \(v\) - speed (km/h), \(t\) - time (h), \(v>0\).
c) Vertical asymptote \(v = 0\), horizontal asymptote \(t = 0\).
d) As speed increases, time decreases (inverse relationship).