Home > PC3 > Chapter 9 > Lesson 9.1.1 > Problem9-13

9-13.

Juan loves to ride his bicycle. Today he is cruising along Exeter Street, which is on flat ground, at $20$ miles per hour ($29.33$ feet per second). On his front wheel is a reflector $11$ inches from the center of the wheel. Juan’s bike has $26$ inch wheels. Assume that the reflector is farthest from the ground when the time $t=0$. Let $h$ be the height of the reflector above the ground (in inches), $t$ be the elapsed time (in seconds) and $x$ the distance (in inches) Juan has traveled in $t$ seconds.

1. How far does Juan travel (in inches) when the wheel makes one complete revolution?

If the wheel has a diameter of $26$ inches, what is the circumference of the wheel?

2. Write an equation for $h(x)$, the height of the reflector, in terms of the distance traveled.

The center of the wheel is $13$ inches off the ground. The reflector oscillates between $11$ inches above and below the reflector.
These hints should tell you the amplitude and vertical shift.

Since the reflector starts at the top, use the cosine function.
Since this function is in terms of distance traveled,  one revolution of the wheel is the same as the circumference you found in part (a), or the period.

3. What is the height of the reflector after Juan has ridden $20$ feet?

Calculate $h(20 \text{ feet})$ using your equation from part (b).
Careful! The equation in part (b) uses inches.

4. Write an equation for $h(t)$, the height of the reflector, in terms of the time $t$.

This will be similar to your equation from part (b), but the period will be the time in seconds to complete one revolution.
Use dimensional analysis to determine "seconds per revolution" for this situation.

5. What is the height of the reflector after Juan has ridden for $5$ minutes?

Calculate $h(5 \text{ minutes})$ using your equation from part (d).
Careful! The equation in part (d) uses time in seconds.