Problem 57 - Precalculus

Question

Refer to the following: The human body temperature normally fluctuates during the day. A person's body temperature can be predicted by the formula \(T=99.1-0.5 \sin \left(x+\frac{\pi}{12}\right),\) where \(x\) is the number of hours since midnight and \(T\) is in degrees Fahrenheit. What is the person's temperature at \(6.00 \mathrm{A} \cdot \mathrm{M} . ?\)

Step-by-Step Solution

Answer

The temperature at 6:00 A.M. is approximately 99.1 degrees Fahrenheit.

Step 1: Identify the Value of \(x\)

Since the question asks for the temperature at 6:00 A.M., we need to determine the value of \(x\). Note that \(x\) represents the number of hours since midnight. Since midnight is 0 hours, 6:00 A.M. is 6 hours after midnight. Therefore, \(x = 6\).

Step 2: Substitute \(x\) in the Formula

Substitute \(x = 6\) into the given formula for temperature, \(T = 99.1 - 0.5 \sin \left(x + \frac{\pi}{12}\right)\). This gives:\[T = 99.1 - 0.5 \sin \left(6 + \frac{\pi}{12}\right)\]

Step 3: Simplify Inside the Sine Function

First, calculate the expression inside the sine function:\[6 + \frac{\pi}{12} = 6 + 0.2618 \approx 6.2618\]

Step 4: Calculate Sine Value

Now, find the sine of the calculated value:\[\sin(6.2618) \approx 0.04428\]

Step 5: Compute the Temperature

Substitute the sine value back into the formula to find the temperature:\[T = 99.1 - 0.5 \times 0.04428\]\[T = 99.1 - 0.02214\]\[T \approx 99.07786\]

Step 6: Final Step: Approximation and Answer

The temperature is typically rounded to one decimal place in practice, so:\[T \approx 99.1 \text{ degrees Fahrenheit.}\]

Key Concepts

Sine Function

The sine function is a fundamental aspect of trigonometry and is commonly used in wave and cycle problems, as it captures oscillations smoothly. In the context of sinusoidal functions like this temperature formula, the sine wave depicts cyclic changes. Here’s how it relates to body temperature:

• The amplitude of the sine function determines the maximum deviation from the average body temperature, which is 99.1°F in this formula.
• The negative coefficient \(-0.5\) indicates that the temperature decreases as the sine value increases, following a downward cycle.
• The \(x + \frac{\pi}{12}\) inside the sine function shifts the wave horizontally by \(\frac{\pi}{12}\) units, which accounts for the time delay in our daily temperature fluctuation cycle.

This cyclical behavior models real-world phenomena such as day-night cycles or seasonal changes in temperatures.
Understanding the sine function allows us to predict temperature changes or any other periodic phenomena effectively without the need for continuous monitoring.

Temperature Calculation

Calculating a person's temperature at a specific time using a trigonometric function involves substituting the correct values into the formula. Here's a breakdown of the temperature calculation process:

• First, identify the number of hours since midnight, as this is represented by \(x\) in the formula. For example, 6:00 A.M. corresponds to \(x = 6\).
• Next, substitute this value into the equation: \ T = 99.1 - 0.5 \sin(x + \frac{\pi}{12}) \ and calculate the expression within the sine function.
• After computing \(x + \frac{\pi}{12}\), which results in approximately \6.2618\, use your calculator to find \sin(6.2618)\, which results in about \0.04428\.

This step-by-step process simplifies finding the temperature at any given time. By rounding off the result to one decimal place, we obtain the most practical and meaningful answer, which is 99.1°F in this example.

Substitution Method

The substitution method is a crucial strategy in algebra and trigonometry, used to replace variables with specific values to solve equations. This method simplifies complicated expressions and makes them manageable:

• Begin with identifying the variable involved in the problem. In our exercise, \(x\) represents the hours since midnight.
• Substitute the determined value directly into the function to solve for the desired quantity. Here, \(x = 6\) is substituted into the temperature function.
• After substitution, work through the operations methodically, ensuring each step is computed accurately, such as finding the sine of the resultant angle.

The substitution method is straightforward and efficient, especially when dealing with trigonometric functions, as it reduces errors and ensures clarity. It’s extremely valuable for checking your work at each manipulation stage of an expression or formula.