A researcher observes and records the height of a weight …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

A researcher observes and records the height of a weight moving up and down on the end of a spring. At the beginning of the observation the weight was at its highest point. From its resting position, it takes 12 seconds for the weight to reach its highest position, fall to its lowest position, and return to its resting position. The difference between the lowest and the highest points is 10 in. Assume the resting position is at y = 0. Use the sine tool to graph the function. The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point.

Short Answer

To graph the height of a weight on a spring, first understand the sine wave equation y = A sin(Bx + C) + D. Then, determine key values: the amplitude (A) is 5 inches, the period is 12 seconds (leading to B = 2œA/12), and D is 0. The final equation for the height over time is y = 5 sin((2œA/12)t).

Step-by-Step Solution

To graph the function representing the height of the weight on a spring, follow these three steps:

Step 1: Understand the Sine Wave Equation

To begin, familiarize yourself with the sine wave equation y = A sin(Bx + C) + D. In this equation:

  • A is the amplitude, which indicates the peak height from the midline.
  • B determines the horizontal stretch or compression based on the period.
  • C represents the phase shift, and D reflects vertical displacement.

Step 2: Determine Key Values

Identify the important parameters for your specific function. For the weight moving on the spring:

  • The amplitude (A) is 5 inches, indicating the maximum height from the resting position.
  • The period is 12 seconds, allowing you to compute B: use B = 2≈ìA/period.
  • The vertical displacement (D) is 0 (since the resting position is y = 0).

Step 3: Write the Final Equation

Now that you have the necessary values, plug them into the sine wave equation. The procedure is straightforward:

  • Substitute 5 for A in the equation.
  • Calculate B using the period: B = 2≈ìA/12.
  • Since there is no phase shift, set C to 0, and D remains 0.
  • Hence, the final equation for the height of the weight as a function of time is y = 5 sin((2≈ìA/12)t).

Related Concepts

Sine Wave Equation

A mathematical representation of periodic oscillations, typically expressed as y = a sin(bx + c) + d, where a, b, c, and d are constants that modify the wave’s amplitude, frequency, phase, and vertical shift, respectively.

Amplitude

The maximum extent of a wave measured from its mean position, indicating how far the wave reaches from its midline (y = 0).

Period

The duration it takes for one complete cycle of a wave; in the context of sine waves, it helps determine the frequency and is used to calculate the value of b in the sine wave equation.

Table Of Contents
  1. A researcher observes and records the height of a weight moving up and down on the end of a spring. At the beginning of the observation the weight was at its highest point. From its resting position, it takes 12 seconds for the weight to reach its highest position, fall to its lowest position, and return to its resting position. The difference between the lowest and the highest points is 10 in. Assume the resting position is at y = 0. Use the sine tool to graph the function. The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point.

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