7. a student rides his bicycle to school each day. the pedals are mounted on a bracket whose center is 29 cm…

7. a student rides his bicycle to school each day. the pedals are mounted on a bracket whose center is 29 cm above the ground. each pedal is 16.5 cm from the center of the bracket. assume that the bicycle is pedaled at 12 cycles per minute. with the starting position of the pedals in a horizontal position at t = 0. a. label the graph with time (seconds) on the x - axis. each line will represent one second. b. label the y - axis as height (cm) from the ground. each line will represent 5 cm. c. sketch the graph of this sinusoidal function for the first three cycles. assume the pedal you are measuring goes “up” on the first rotation. d. write the equation of the sine curve with the time given in seconds. e. calculator active. when is the pedal 40 cm above the ground for the 3rd time? f. calculator active. how high is the pedal after 23 seconds?

7. a student rides his bicycle to school each day. the pedals are mounted on a bracket whose center is 29 cm above the ground. each pedal is 16.5 cm from the center of the bracket. assume that the bicycle is pedaled at 12 cycles per minute. with the starting position of the pedals in a horizontal position at t = 0. a. label the graph with time (seconds) on the x - axis. each line will represent one second. b. label the y - axis as height (cm) from the ground. each line will represent 5 cm. c. sketch the graph of this sinusoidal function for the first three cycles. assume the pedal you are measuring goes “up” on the first rotation. d. write the equation of the sine curve with the time given in seconds. e. calculator active. when is the pedal 40 cm above the ground for the 3rd time? f. calculator active. how high is the pedal after 23 seconds?

Answer

Explanation:

Step1: Find the period

The bicycle is pedaled at 12 cycles per minute. So the period $T$ in seconds is $\frac{60}{12}=5$ seconds.

Step2: Determine the amplitude

The center of the bracket is 29 cm above the ground and each pedal is 16.5 cm from the center of the bracket. So the amplitude $A = 16.5$.

Step3: Find the vertical - shift

The vertical - shift $D=29$.

Step4: Determine the phase - shift

Since the starting position of the pedals is in a horizontal position at $t = 0$ and the pedal goes "up" on the first rotation, the phase - shift $C = 0$.

Step5: Write the sine function

The general form of a sine function is $y=A\sin(B(t - C))+D$. We know that $B=\frac{2\pi}{T}$, and since $T = 5$, $B=\frac{2\pi}{5}$. So the equation of the sine curve is $y = 16.5\sin(\frac{2\pi}{5}t)+29$.

Step6: Solve for when $y = 40$ for the 3rd time

Set $y = 40$, so $40=16.5\sin(\frac{2\pi}{5}t)+29$. Then $11 = 16.5\sin(\frac{2\pi}{5}t)$, and $\sin(\frac{2\pi}{5}t)=\frac{11}{16.5}=\frac{2}{3}$. The general solution for $\sin(x)=\frac{2}{3}$ is $x=\sin^{- 1}(\frac{2}{3})+2k\pi$ or $x=\pi-\sin^{- 1}(\frac{2}{3})+2k\pi$. For the sine function $y = 16.5\sin(\frac{2\pi}{5}t)+29$, we want the 3rd positive solution. First, $\frac{2\pi}{5}t=\sin^{- 1}(\frac{2}{3})+2\pi$ (the first positive solution for our context). Solving for $t$, we get $t=\frac{5}{2\pi}(\sin^{- 1}(\frac{2}{3})+2\pi)$. The second positive solution: $\frac{2\pi}{5}t=\pi-\sin^{- 1}(\frac{2}{3})+2\pi$. The third positive solution: $\frac{2\pi}{5}t=\sin^{- 1}(\frac{2}{3}) + 4\pi$. Then $t=\frac{5}{2\pi}(\sin^{- 1}(\frac{2}{3})+4\pi)\approx10.4$ seconds.

Step7: Find the height at $t = 23$ seconds

Substitute $t = 23$ into $y = 16.5\sin(\frac{2\pi}{5}t)+29$. $\frac{2\pi}{5}\times23=\frac{46\pi}{5}=9\pi+\frac{\pi}{5}$. $\sin(\frac{46\pi}{5})=\sin(9\pi+\frac{\pi}{5})=-\sin(\frac{\pi}{5})$. $y = 16.5\times(-\sin(\frac{\pi}{5}))+29\approx16.5\times(- 0.5878)+29\approx19.3$ cm.

Answer:

a. and b. and c. (Sketching and labeling are visual tasks. For the graph: x - axis is labeled with time in seconds (0 - 15 for 3 cycles as period is 5 seconds), y - axis is labeled with height in cm (ranging from about $29 - 16.5=12.5$ to $29 + 16.5 = 45.5$ with each line representing 5 cm). The sine - curve starts at $(0,29)$ and goes up first). d. $y = 16.5\sin(\frac{2\pi}{5}t)+29$ e. Approximately 10.4 seconds f. Approximately 19.3 cm