I do not see why I got two different equations for my answer

A weight attached to the end of along spring is bounicng up and down. As it bounces, its distance from the floor varies sinusoidally with time. You start a stopwatch. When the stopwatch reads 0.3 s, the weight first reaches a height point 60 cm above the floor. THe next low point, 40 cm above the floor, occurs at 1.8 s.

Write an equation expressing distance from the floor in terms of the number of seconds the stopwatch reads.

I got two answers and don't see why

50 + 10sin 1.5^-1(x+.45)pi

50 + -10 sin -1.5^-1 (x - 2.55)pi

They are different by 3 PI. Think about the sine function with a difference of 3PI, it changes sign. So, it is right, as if you add 3 pi, the sign changes, they are the SAME answer.

To understand why you got two different equations for your answer, let's break down the problem. We are given the following information:

1. At 0.3 seconds, the weight reaches a high point 60 cm above the floor.
2. At 1.8 seconds, the weight reaches a low point 40 cm above the floor.

We need to write an equation expressing the distance from the floor in terms of the number of seconds the stopwatch reads. Since the motion is described as varying sinusoidally with time, we can use a sine function to model the height of the weight.

The general form of a sine function is: f(x) = A * sin(Bx + C) + D

Where:
- A is the amplitude (the maximum distance from the mean/average value)
- B is the frequency (related to the period of the function)
- C is the phase shift (horizontal displacement)
- D is the vertical shift (mean value)

Based on the given information, let's determine the values for A, B, C, and D.

1. Amplitude (A):
The amplitude of the oscillation is half the difference between the high point and the low point. In this case, the high point is 60 cm and the low point is 40 cm. The amplitude can be calculated as: A = (60 - 40) / 2 = 10 cm.

2. Frequency and Period (B):
The frequency is related to the period of the function. The period is the time it takes for one complete cycle of the oscillation. In this case, the time between the high point and the next low point is 1.8 seconds - 0.3 seconds = 1.5 seconds. The frequency is the reciprocal of the period: B = 1 / 1.5 = 2/3.

3. Phase Shift (C):
The phase shift is related to the horizontal displacement of the function. It represents any shift to the right or left. In this case, at 0.3 seconds, the weight reaches a high point. The phase shift can be calculated as the opposite of the time value at the high point: C = -(0.3) = -0.3.

4. Vertical Shift (D):
The vertical shift is related to the mean/average value of the function. In this case, the average height is halfway between the high and low points, which is 50 cm. Therefore, D = 50.

Now, let's substitute the calculated values into the general sine function equation:

f(x) = 10 * sin((2/3)x - 0.3) + 50

Simplifying the equation, we get:

f(x) = 10 * sin((2/3)x - 0.3) + 50

So, the equation expressing the distance from the floor in terms of the number of seconds the stopwatch reads is f(x) = 10 * sin((2/3)x - 0.3) + 50.

If you obtained different equations, then there might have been errors in the calculations or misunderstandings of the given information. Make sure to double-check your work and follow the step-by-step process outlined above to arrive at the correct answer.