Question

b. are there one or more values of $t$ at which the equation $a(t) = b(t)$ is true? if so, which one or which ones?

c. identify at least two values of $t$ at which the inequality $b(t) < a(t)$ is true.

1. john and his friends are playing hide-and-seek at recess. everyone starts at the base. once the seeker starts counting, john runs straight ahead until he finds a good hiding spot behind the slide. he hides there for a short time until the seeker goes looking in a different direction. then, john runs back to the base.

sketch a graph could show johns distance from the base over time.

Explanation:

1. Axes Setup: The x-axis represents time ($t$) and the y-axis represents distance from the base. Start at the origin $(0,0)$ since John begins at the base at time 0.
2. First Segment: John runs straight away from the base, so distance increases linearly with time. Draw a rising straight line from $(0,0)$ to a point $(t_1, d_1)$ where $t_1$ is the time he reaches the hiding spot, and $d_1$ is his maximum distance from the base.
3. Second Segment: John hides, so his distance from the base stays constant as time passes. Draw a horizontal straight line from $(t_1, d_1)$ to a point $(t_2, d_1)$ where $t_2 - t_1$ is the time he spends hiding.
4. Third Segment: John runs back to the base, so distance decreases linearly with time until it reaches 0. Draw a falling straight line from $(t_2, d_1)$ to the point $(t_3, 0)$ where $t_3$ is the time he arrives back at the base.

Answer:

A correct graph will have three distinct segments:

1. A positive-sloped line starting at $(0,0)$ (moving away from base)
2. A horizontal line (stationary/hiding)
3. A negative-sloped line ending at a point on the x-axis (returning to base)

(Note: The provided grid sketch already matches this structure, with a rising line, flat line, and falling line, which is the correct representation.)