Bot GPT 4-o mini To determine whether to use a sine or cosine function for Cheryl's heart on the bike tire, we note the following:
-
Amplitude: The heart has an amplitude of 12. This means that the heart will reach a maximum height of 12 units above and below the midline.
-
Midline (D): The midline is given as \(D = 12\). This indicates that the vertical midpoint of the function is at \(y = 12\).
-
Revolutions per minute: The bike tire makes 70 revolutions per minute.
-
Period: The period of a function in general can be determined by the formula: \[ \text{Period} = \frac{60}{\text{revolutions per minute}} \times 2\pi \] Substituting the values: \[ \text{Period} = \frac{60}{70} \times 2\pi \approx 5.366 \text{ seconds} \]
-
Phase shift: There is no phase shift specified, so we will consider it to be 0.
Choosing Sine or Cosine
Since the heart starts at the midline and goes upward (if we visualize it), a sine function is usually the best fit given that sine starts at (0, 0) and then goes up.
Writing the Function
The general form for a sine function with the given parameters is: \[ y(t) = A \sin(B(t - C)) + D \]
Where:
- \(A\) is the amplitude; \(A = 12\).
- \(B\) relates to the period; \(B = \frac{2\pi}{P}\).
- \(C\) is the phase shift; here, \(C = 0\).
- \(D\) is the midline; \(D = 12\).
Calculating \(B\): \[ B = \frac{2\pi}{\text{Period}} \approx \frac{2\pi}{5.366} \approx 1.169 \]
Putting it all together, the function would be: \[ y(t) = 12 \sin\left(1.169t\right) + 12 \]
This sine function models the position of the heart on the bike tire as it rotates, rising and falling with an amplitude of 12 above and below the midline of 12, completing a full cycle in approximately 5.366 seconds.
1 answer