The pressure of a fluid flowing through a closed system, , after seconds can be modeled by the function . Answer parts (a) through © below.
(a) In the interval , determine the times at which the pressure of the fluid is 100 mmHg .
The solution set is .
(Type an integer or decimal rounded to two decimal places as needed. Use a comma to separate answers as needed.)
(b) In the interval , determine the times at which the pressure of the fluid is 140 mmHg .
The solution set is .
(Type an integer or decimal rounded to two decimal places as needed. Use a comma to separate answers as needed.)
© In the interval , determine the times at which the pressure of the fluid is between 100 and 105 mmHg .

Ask by Carrillo Norton. in the United States

Jan 25,2025

Question

The pressure of a fluid flowing through a closed system, , after seconds can be modeled by the function . Answer parts (a) through © below.
(a) In the interval , determine the times at which the pressure of the fluid is 100 mmHg .
The solution set is .
(Type an integer or decimal rounded to two decimal places as needed. Use a comma to separate answers as needed.)
(b) In the interval , determine the times at which the pressure of the fluid is 140 mmHg .
The solution set is .
(Type an integer or decimal rounded to two decimal places as needed. Use a comma to separate answers as needed.)
© In the interval , determine the times at which the pressure of the fluid is between 100 and 105 mmHg .

Ask by Carrillo Norton. in the United States

Jan 25,2025

UpStudy AI · Accepted Answer

To determine the times $$ t $$ in the interval $$[0, 1]$$ seconds where the pressure $$ P(t) $$ is between 100 mmHg and 105 mmHg, we'll follow these steps:

### Given:
$$ P(t) = 100 + 40 \sin\left(\frac{12\pi}{5} t\right) $$

We want to find $$ t $$ such that:
$$ 100 < P(t) < 105 $$
$$ 0 < 40 \sin\left(\frac{12\pi}{5} t\right) < 5 $$
$$ 0 < \sin\left(\frac{12\pi}{5} t\right) < \frac{1}{8} $$

### Solving for $$ t $$:

1. **Find the general solutions where $$ \sin\left(\frac{12\pi}{5} t\right) = \frac{1}{8} $$:**
   
   $$ \frac{12\pi}{5} t = \arcsin\left(\frac{1}{8}\right) $$
   
   $$ t = \frac{5}{12\pi} \arcsin\left(\frac{1}{8}\right) \approx 0.0167 \text{ seconds} $$
   
   Similarly, considering the sine function's periodicity:
   
   $$ t \approx \frac{5}{12\pi} (\pi - \arcsin\left(\frac{1}{8}\right)) \approx 0.40 \text{ seconds} $$
   
   $$ t \approx \frac{5}{12\pi} (2\pi) \approx 0.83 \text{ seconds} $$
   
   $$ t \approx \frac{5}{12\pi} (2\pi + \arcsin\left(\frac{1}{8}\right)) \approx 0.85 \text{ seconds} $$

2. **Determine the intervals where $$ \sin\left(\frac{12\pi}{5} t\right) $$ is between 0 and $$ \frac{1}{8} $$:**
   
   - **First Interval:** $$ t $$ from 0 to approximately 0.02 seconds.
   
   - **Second Interval:** $$ t $$ from approximately 0.40 to 0.42 seconds.
   
   - **Third Interval:** $$ t $$ from approximately 0.83 to 0.85 seconds.

### **Solution Set:**
$$ \{ [0, 0.02], [0.40, 0.42], [0.83, 0.85] \} $$

**Rounded to two decimal places:**
$$ \{ [0.00, 0.02], [0.40, 0.42], [0.83, 0.85] \} $$

### **Final Answer:**
The pressure is between 100 and 105 mmHg for all times t in [0.00, 0.02], [0.40, 0.42], and [0.83, 0.85] seconds.
The pressure is between 100 and 105 mmHg at times approximately 0.00, 0.02, 0.40, 0.42, 0.83, and 0.85 seconds.

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