Question

1. isaac lives 3 miles away from his school. school ended at 3 pm and isaac began his walk home with his friend tate who lives 1 mile away from the school, in the direction of isaacs house. isaac stayed at tates house for a while and then started home. on the way he stopped at the library. then he hurried home. the graph at the right is a piece - wise defined function that shows isaacs distance from home during the time it took him to arrive home.

a. how much time passed between school ending and isaacs arrival home?
b. how long did isaac stay at tates house?
c. how far is the library from isaacs house?
d. where was isaac, 3 hours after school ended?
e. use function notation to write a mathematical expression that says the same thing as question d.
f. when was isaac walking the fastest? how fast was he walking?
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Explanation:

Step1: Analyze the graph for time passed between school ending and arrival home

The school ended at \(3\) pm (time \(0\) on the graph). He arrived home at \(2.5\) hours on the graph. So the time passed is \(2.5\) hours.

Step2: Analyze the graph for time at Tate's house

The flat - part of the graph (where distance doesn't change) represents the time at Tate's house. The start of the flat - part is at \(1.5\) hours and the end is at \(2\) hours. So the time is \(2 - 1.5=0.5\) hours.

Step3: Analyze the graph for distance of library from house

The distance at the library stop is \(0.5\) miles (from the \(y\) - axis).

Step4: Analyze the graph for time \(3\) hours after school ended

\(3\) hours after school ended (time \(3\) on the graph). The distance from home is \(0\) miles (he is home).

Step5: Write the function for question d

Let \(t\) be the time (in hours) after school ended and \(d(t)\) be the distance from home. For example, if we consider the first non - flat segment (going to the library): from \(t = 0\) to \(t = 1\), \(d(t)=3 - t\) (since he starts \(3\) miles from home and moves towards home at a rate of \(1\) mile per hour). Then from \(t = 1\) to \(t = 1.5\), \(d(t)=2\) (distance remains \(2\) miles from home). From \(t = 1.5\) to \(t = 2\), \(d(t)=2\) (stays at Tate's). From \(t = 2\) to \(t = 2.5\), \(d(t)=2-(t - 2)\times4\) (because the slope is \(- 4\), \(d(t)=10 - 4t\)).

Step6: Analyze the slope for fastest walking

The slope of the distance - time graph represents speed. The steeper the slope (in the negative direction for going home), the faster the speed. The segment from \(t = 2\) to \(t = 2.5\) has a slope of \(\frac{0 - 2}{2.5 - 2}=\frac{- 2}{0.5}=-4\) (speed of \(4\) miles per hour). The first segment (from \(t = 0\) to \(t = 1\)) has a slope of \(\frac{2 - 3}{1-0}=-1\) (speed of \(1\) mile per hour).

Answer:

a. \(2.5\) hours; b. \(0.5\) hours; c. \(0.5\) miles; d. \(d(t)=

$$\begin{cases}3 - t, &0\leq t\lt1\\2, &1\leq t\lt1.5\\2, &1.5\leq t\lt2\\10 - 4t, &2\leq t\leq2.5\end{cases}$$

\); e. As shown above; f. The segment from \(t = 2\) to \(t = 2.5\) (speed of \(4\) miles per hour)