Question

for each scenario below, choose the best graph. (a) reuben bikes from home to work. reuben’s distance to work time 0 x y reuben’s distance to work time 0 x y reuben’s distance to work time 0 x y reuben’s distance to work time 0 x y (b) ryan rides his bike at a constant speed. ryan’s speed time 0 x y ryan’s speed time 0 x y ryan’s speed time 0 x y ryan’s speed time 0 x y

Explanation:

Part (a) - Reuben’s Bike Ride to Work

Step1: Analyze Distance - Time Relationship

When Reuben bikes from home to work, the distance to work should increase as time (x - axis) increases (since he's moving towards work). A constant speed means distance \( d = vt \) (linear relationship, positive slope).

• First graph: Horizontal line (distance doesn't change with time) → wrong.
• Second graph: Vertical line (time doesn't change, impossible) → wrong.
• Third graph: Line from (0,0) with positive slope → distance increases with time (correct for moving at constant speed).
• Fourth graph: Starts after time > 0, but he starts at time 0 → wrong.

Step2: Confirm the Correct Graph

The third graph (bottom - most in Reuben’s section) shows distance to work increasing linearly with time, matching a constant - speed bike ride from home (starts at distance 0, time 0).

Part (b) - Ryan’s Constant - Speed Bike Ride

Step1: Analyze Speed - Time Relationship

For constant speed, speed (y - axis) should not change with time (x - axis). So the graph of speed vs. time should be a horizontal line (slope = 0).

• First graph: Horizontal line (speed constant) → correct candidate.
• Second graph: Vertical line (speed changes infinitely with time) → wrong.
• Third graph: Line with positive slope (speed increasing) → wrong.
• Fourth graph: Line with positive slope (speed increasing after some time) → wrong.

Step2: Confirm the Correct Graph

The first graph (top - most in Ryan’s section) has a horizontal line, meaning speed is constant over time, which matches "rides at a constant speed".

Answer:

(a) The third graph in Reuben’s “Distance to Work” section (the one with a line from (0,0) with positive slope).
(b) The first graph in Ryan’s “Speed” section (the horizontal line graph).