T(t) = – 9.5sin(t) - + 77.5 Now that we have our function, we can answer the question at hand. How many hours after midnight does the temperature first reach 84°? Since 84 > 77.5, we know that our answer will be between 12 and 18 because our function will be less than 77.5 until 12, and then will be increasing until it reaches its maximum at 18. Solve for t when T(t) = 84. T(t) = 84 = - 9.5 sin| + 77.5 84 – 77.5 = - 9.5 sin| 84 – 77.5 sin ) = sin -9.5 77.5 – 84 9.5 12 = 14.878 t = arcsin So, the temperature will first reach 84° 14.878 hours after midnight,

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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I am not understanding what they did to obtain that result (in the image). 

My question is not about the actual exercise. My question is about the las 2 steps of the solution. How did they solve fot t? 

**Temperature Function Analysis**

Given the temperature function:
\[ T(t) = -9.5 \sin \left( \frac{\pi}{12} t \right) + 77.5 \]

**Objective:**
Determine how many hours after midnight the temperature first reaches 84°.

Since 84° is greater than 77.5°, our answer will fall between 12 and 18 hours because the function \( T(t) \) is less than 77.5 up to 12 hours and then starts increasing, peaking at 18 hours.

**Steps to Solve for \( t \) when \( T(t) = 84 \):**

1. Set the equation:
\[ 84 = -9.5 \sin \left( \frac{\pi}{12} t \right) + 77.5 \]

2. Subtract 77.5 from both sides:
\[ 84 - 77.5 = -9.5 \sin \left( \frac{\pi}{12} t \right) \]
\[ 6.5 = -9.5 \sin \left( \frac{\pi}{12} t \right) \]

3. Divide both sides by -9.5:
\[ \frac{6.5}{-9.5} = \sin \left( \frac{\pi}{12} t \right) \]

4. Calculate the left-hand side:
\[ \sin \left( \frac{\pi}{12} t \right) = -\frac{6.5}{9.5} \]

5. Take the arcsine (inverse sine) of both sides to solve for \( t \):
\[ \frac{\pi}{12} t = \arcsin \left( -\frac{6.5}{9.5} \right) \]

6. Divide by \( \frac{\pi}{12} \) to isolate \( t \):
\[ t = \arcsin \left( -\frac{6.5}{9.5} \right) \times \frac{12}{\pi} \]

7. Solve for \( t \):
\[ t \approx 14.878 \]

**Conclusion:**
The temperature first reaches 84° approximately 14.878 hours after midnight.
Transcribed Image Text:**Temperature Function Analysis** Given the temperature function: \[ T(t) = -9.5 \sin \left( \frac{\pi}{12} t \right) + 77.5 \] **Objective:** Determine how many hours after midnight the temperature first reaches 84°. Since 84° is greater than 77.5°, our answer will fall between 12 and 18 hours because the function \( T(t) \) is less than 77.5 up to 12 hours and then starts increasing, peaking at 18 hours. **Steps to Solve for \( t \) when \( T(t) = 84 \):** 1. Set the equation: \[ 84 = -9.5 \sin \left( \frac{\pi}{12} t \right) + 77.5 \] 2. Subtract 77.5 from both sides: \[ 84 - 77.5 = -9.5 \sin \left( \frac{\pi}{12} t \right) \] \[ 6.5 = -9.5 \sin \left( \frac{\pi}{12} t \right) \] 3. Divide both sides by -9.5: \[ \frac{6.5}{-9.5} = \sin \left( \frac{\pi}{12} t \right) \] 4. Calculate the left-hand side: \[ \sin \left( \frac{\pi}{12} t \right) = -\frac{6.5}{9.5} \] 5. Take the arcsine (inverse sine) of both sides to solve for \( t \): \[ \frac{\pi}{12} t = \arcsin \left( -\frac{6.5}{9.5} \right) \] 6. Divide by \( \frac{\pi}{12} \) to isolate \( t \): \[ t = \arcsin \left( -\frac{6.5}{9.5} \right) \times \frac{12}{\pi} \] 7. Solve for \( t \): \[ t \approx 14.878 \] **Conclusion:** The temperature first reaches 84° approximately 14.878 hours after midnight.
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