Problem 58 - Precalculus with Calculus Previews

Question

Fahrenheit Temperature Suppose that $$ T(t)=50+10 \sin \frac{\pi}{12}(t-8) $$ \(0 \leq t \leq 24\), is a mathematical model of the Fahrenheit temperature at \(t\) hours after midnight on a certain day of the week. (a) What is the temperature at \(8 \mathrm{~A} . \mathrm{M}\).? (b) At what time(s) does \(T(t)=60\) ? (c) Sketch the graph of \(T\). (d) Find the maximum and minimum temperatures and the times at which they occur.

Step-by-Step Solution

Answer

(a) 50°F at 8 A.M.; (b) 2 P.M.; (c) See sketch steps; (d) Max: 60°F at 2 P.M., Min: 40°F at 2 A.M.

Step 1: Evaluating Temperature at 8 A.M.

First, we need to determine the temperature at 8 A.M., which corresponds to \( t = 8 \). Substitute \( t = 8 \) into the equation: \[ T(8) = 50 + 10 \sin \frac{\pi}{12}(8-8) = 50 + 10 \times 0 = 50 \]So, the temperature at 8 A.M. is 50°F.

Step 2: Finding Times When Temperature is 60°F

We need to solve for \( t \) when \( T(t) = 60 \):\[ 60 = 50 + 10 \sin \frac{\pi}{12}(t-8) \]Subtract 50 from both sides:\[ 10 = 10 \sin \frac{\pi}{12}(t-8) \]Divide by 10:\[ 1 = \sin \frac{\pi}{12}(t-8) \]Since \( \sin \theta = 1 \) at \( \theta = \frac{\pi}{2} + 2k\pi \), we have:\[ \frac{\pi}{12}(t-8) = \frac{\pi}{2} + 2k\pi \]Solving for \( t \), we get:\[ t = 14 + 24k \]Considering \( 0 \leq t \leq 24 \), we find \( t = 14 \).Thus, the times when \( T(t) = 60 \) are 2 P.M. (14:00).

Step 3: Sketching the Temperature Graph

The function \( T(t) = 50 + 10 \sin \frac{\pi}{12}(t-8) \) is a sinusoidal function centered at 50°F with an amplitude of 10. The period of this sine wave can be calculated as:\[ P = \frac{2\pi}{\frac{\pi}{12}} = 24 \]This indicates a complete cycle is 24 hours. The graph oscillates between 40°F and 60°F.

Step 4: Finding Maximum and Minimum Temperatures

The maximum temperature corresponds to the maximum value of the function \( 10 \sin \theta \), which happens when \( \sin \theta = 1 \). At maximum, \( T(t) = 50 + 10 = 60 \). This occurs when \( t = 14 \) as calculated previously.The minimum temperature corresponds to the minimum value of \( 10 \sin \theta \), which happens when \( \sin \theta = -1 \). At minimum, \( T(t) = 50 - 10 = 40 \). Solving for \( t \), \( \frac{\pi}{12}(t-8) = \frac{3\pi}{2} + 2m\pi \) gives \( t = 2 \) (2 A.M.).Therefore, the minimum temperature of 40°F occurs at 2 A.M., and the maximum temperature of 60°F occurs at 2 P.M.

Key Concepts

Sinusoidal Functions

Sinusoidal functions are a type of periodic function that can represent oscillating phenomena, such as sound waves, tides, and even temperature changes. They are called "sinusoidal" because they are based on the sine function, one of the fundamental trigonometric functions. The general form of a sinusoidal function is given by:

• \( y = A \sin(Bx - C) + D \)

In this formula:

• \( A \) is the amplitude, indicating how high and low the wave goes from the center line.
• \( B \) affects the period, which is the length of one complete wave cycle.
• \( C \) is the phase shift, determining the horizontal translation.
• \( D \) is the vertical shift, which moves the wave up or down.

These functions are useful in many practical applications. Since they repeat at regular intervals, they can model any phenomena that follows a pattern over time. In the context of our exercise, the temperature throughout the day can be precisely modeled using a sinusoidal function as it naturally oscillates between maximum and minimum values.
It is important to understand these parameters when working with sinusoidal functions, as they let us tailor the function to match real-world situations with precision.

Temperature Modeling

Modeling temperature using sinusoidal functions is a practical application of mathematics. In real life, temperatures vary predictably over time, often rising and falling over a 24-hour period. This cyclical pattern makes

• Sinusoidal functions the perfect choice to represent these changes.

In our exercise:

• The temperature function \( T(t) = 50 + 10 \sin \frac{\pi}{12}(t-8) \) gives a model of temperature over 24 hours.
• The base temperature is 50°F, and the amplitude, 10°F, indicates the maximum variation above and below the base.
• The periodicity is evident as the cycle completes every 24 hours."

This function allows us to predict temperatures at different times and understand the temperature dynamics throughout the day. Temperature modeling like this is extremely useful in applications like agriculture, energy consumption, and climate studies, where precise temperature information is crucial for decision-making.
The simplicity and efficiency of using sinusoidal functions for temperature modeling lie in their natural fit to describe periodic natural phenomena.

Maximum and Minimum Values

Finding the maximum and minimum values of a sinusoidal function is essential to understand the full range of a modeled phenomenon. In our example, the maximum and minimum temperatures happen when the sine component of the function reaches its highest and lowest possible values, which are 1 and -1 respectively.

• The maximum value of the function occurs when \( \sin(\theta) = 1 \). With the equation \( T(t) = 50 + 10 \sin \left(\frac{\pi}{12}(t-8)\right) \), the maximum temperature is \( 60°F \), happening at 2 P.M. (\( t = 14 \)).
• The minimum value occurs when \( \sin(\theta) = -1 \). Thus, the minimum temperature is \( 40°F \), occurring at 2 A.M. (\( t = 2 \)).

These calculations help in determining the extent to which the temperature fluctuates during a 24-hour period.
The understanding of such maximum and minimum points is pertinent in many fields. For instance, knowing the coldest time of the night or the warmest time of the day is useful in activities like planning thermal energy use or scheduling agricultural tasks.