Question

The pressure P (in pounds per square foot), in a pipe varies over time. Three times an hour, the pressure oscillates from a low of 100 to a high of 200 and then back to a low of 100. The pressure at time t=0 is 100. Let the function P=f(t) denote the pressure in pipe at time t minutes. Find a possible formula for the function P=f(t) described above.

This is the problem that I am trying to solve and so far I came up with 50sin((2pi/20)t)+150. 50 is the amplitude, 150 is the midline, and 20 is the period. I put the components together but for some reason it's incorrect and I would like to know why an find the actual answer.

Answer 1

The formula you have provided, 50sin((2pi/20)t) + 150, is close to the correct form but has a slight error in the period. Let's analyze the problem to find the correct formula.

We are given that the pressure oscillates from a low of 100 to a high of 200 and then back to a low of 100 three times per hour. This means there are a total of six oscillations per hour, or one oscillation every 10 minutes since there are 60 minutes in an hour.

The midline or average pressure is 150, and the amplitude is half the difference between the maximum and minimum values, which is (200 - 100)/2 = 50.

Using this information, we can write the correct formula for the function P=f(t) as:

P = 50sin((2pi/10)t) + 150

In this formula, the period is 10 minutes (not 20 as in your initial attempt). The function correctly models the pressure oscillating between 100 and 200 three times per hour, with a midline at 150.

Answer 2

Your approach is close, but there are a few errors in the formula you provided. Let's analyze the problem and find the correct formula step by step.

Given:

- The pressure oscillates three times per hour.

- The pressure starts at a low of 100 and reaches a high of 200 before returning to a low of 100.

- The pressure at time t=0 is 100.

To determine the formula for the function P=f(t), we can break it down into several components:

1. Amplitude: The amplitude represents half the difference between the maximum and minimum values. In this case, the maximum value is 200 and the minimum value is 100. So the amplitude is (200 - 100) / 2 = 50.

2. Midline: The midline represents the average of the maximum and minimum values. In this case, the average is (200 + 100) / 2 = 150.

3. Period: The period is the time it takes for one complete oscillation. In this case, we know that there are three oscillations per hour. Since there are 60 minutes in an hour, the period is 60 / 3 = 20 minutes.

Based on these components, we can construct the formula for the function P=f(t) as follows:

P = 50sin((2π/20)t) + 150

Notice that the frequency (2π/20) represents the number of oscillations per unit of time (t), and multiplying it by t gives us the phase shift. By adding the midline (150) to the amplitude times the sine function, we obtain the desired pressure function.

Therefore, the correct formula for the function P=f(t) is:

P = 50sin((2π/20)t) + 150