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Question 2

A. Predict the height of the heart exactly 30 seconds into the ride. Use your equation from Question 1. Show your work.
B. Predict the height of the heart exactly 5 minutes into the ride. Use your equation from Question 1. Show your work.
Question 1 Cheryl’s cousin is about to embark on a 50-mile bike ride. Cheryl draws a little heart on the front wheel of her cousin’s bike for good luck. When Cheryl’s cousin starts riding, the heart goes up before coming back down and hitting the pavement. The diameter of the wheel is [choose a number between 20 – 27.5] inches. Cheryl’s cousin rides at a steady pace and the wheel makes a frequency of [choose a number between 60 – 80] revolutions per minute. Find the equation that models the height of the heart in inches as a function of time (x) in minutes

1 answer

To solve this problem, we first need to establish a model for the height of the heart based on the diameter of the bike wheel and the frequency of its revolutions.

Assumptions:

  1. Since a bike wheel is circular, the motion of the heart can be represented as a sinusoidal function.
  2. The heart's height will oscillate between 0 inches (the low point, when it touches the ground) and the diameter of the wheel (the high point, when it is at the top of the wheel).

Modeling the Heart's Height:

  1. Choose the diameter and frequency: Let's choose:

    • Diameter of the wheel (D) = 24 inches (this is between 20 and 27.5)
    • Frequency (f) = 70 revolutions per minute (this is between 60 and 80)

  2. Calculate the amplitude and vertical shift:

    • Amplitude (A) = D/2 = 24/2 = 12 inches
    • Vertical shift (B) = D/2 = 12 inches (this shifts the function up so that it oscillates between 0 and 24 inches)

  3. Determine angular frequency (ω):

    • The wheel makes 70 revolutions per minute, which is converted into radians since \(1\) revolution = \(2π\) radians.
    • Thus, \(ω = 70 \text{ rev/min} \times \frac{2π \text{ radians}}{1 \text{ rev}} = 140π \text{ radians/min}\).

  4. Formulate the height equation: Since the heart starts at its maximum height, we can use the cosine function: \[ h(x) = A \cos(ωx) + B \] Substituting in for A, ω, and B: \[ h(x) = 12 \cos(140πx) + 12 \]

A. Predict the height of the heart exactly 30 seconds into the ride.

Convert seconds to minutes: \[ 30 \text{ seconds} = \frac{30}{60} = 0.5 \text{ minutes} \] Now substitute \(x = 0.5\) into the height equation: \[ h(0.5) = 12 \cos(140π(0.5)) + 12 \] Calculating: \[ h(0.5) = 12 \cos(70π) + 12 \] Since \(70π\) corresponds to the cosine function, and knowing that cosine of an odd multiple of π is -1: \[ h(0.5) = 12 \times (-1) + 12 = -12 + 12 = 0 \text{ inches} \]

B. Predict the height of the heart exactly 5 minutes into the ride.

Substitute \(x = 5\) into the height equation: \[ h(5) = 12 \cos(140π(5)) + 12 \] Calculating: \[ h(5) = 12 \cos(700π) + 12 \] According to the periodic nature of the cosine function: \[ \cos(700\pi) = \cos(0) = 1 \] So: \[ h(5) = 12 \times 1 + 12 = 12 + 12 = 24 \text{ inches} \]

Summary of Predictions:

A. The height of the heart exactly 30 seconds into the ride is 0 inches.
B. The height of the heart exactly 5 minutes into the ride is 24 inches.