Problem 72 - Precalculus: Functions and Graphs

Question

The average monthly high temperature \(T\left(\text { in }^{\circ} \mathrm{F}\right)\) in Augusta, Georgia, can be approximated using the function $$ T(t)=17 \cos \left(\frac{\pi}{6} t-\frac{7 \pi}{6}\right)+75 $$ where \(t\) is in months and \(t=1\) corresponds to January. (a) Graph \(T\) over the two-year interval \([1,25]\) (b) Calculate the average high temperature in April and in December. Graphically approximate the months when the average high temperature is \(67^{\circ} \mathrm{F}\) or lower.

Step-by-Step Solution

Answer

April is 75°F, December is 58°F. Temperature is ≤ 67°F around March, October-December.

Step 1: Understand the Function

The given function is \( T(t) = 17 \cos\left(\frac{\pi}{6}t - \frac{7\pi}{6}\right) + 75 \). This is a cosine function with amplitude 17, a vertical shift of 75, and a horizontal shift due to the phase \( -\frac{7\pi}{6} \). It models the oscillating temperature throughout the year.

Step 2: Graph the Function

You need to graph the function over the interval \([1, 25]\), which corresponds to two years of months starting from January (\( t = 1 \)). Use graphing software or a graphical calculator to plot \( T(t) \) from \( t = 1 \) to \( t = 25 \). Ensure you label the x-axis as months and the y-axis as temperature in degrees Fahrenheit.

Step 3: Calculate Temperature in April

April corresponds to \( t = 4 \). Substitute \( t = 4 \) into the function: \[ T(4) = 17 \cos\left(\frac{\pi}{6} \times 4 - \frac{7\pi}{6}\right) + 75 \]. This simplifies to \( T(4) = 17 \cos\left(\frac{\pi}{2}\right) + 75 = 75 \).

Step 4: Calculate Temperature in December

December corresponds to \( t = 12 \). Substitute \( t = 12 \) into the function: \[ T(12) = 17 \cos\left(\frac{\pi}{6} \times 12 - \frac{7\pi}{6}\right) + 75 \]. Simplifying gives \( T(12) = 17 \cos(\pi) + 75 = 75 - 17 = 58 \).

Step 5: Graphically Approximate Months with Temperature \( \leq 67^{\circ} \) F

From the graph derived in Step 2, identify where \( T(t) \leq 67 \). These will be points where the graph dips below or equals the temperature line \( y = 67 \). The graph shows \( T(t) \leq 67 \) approximately around March (\( t \approx 3 \)), October to December (\( 10 \leq t \leq 12 \)), and similar months in the second year.

Key Concepts

Amplitude

The amplitude of a trigonometric function, like a cosine or sine function, defines its maximum and minimum values. For the function in our exercise, \( T(t) = 17 \cos\left(\frac{\pi}{6}t - \frac{7\pi}{6}\right) + 75 \), the amplitude is given by the coefficient of the cosine term, which is 17. This tells us how far the peaks of the function are from the midpoint or average value.
This means the temperature swings 17 degrees Fahrenheit above and below the average, or centerline, temperature of the function. Knowing the amplitude helps us understand the extent of temperature variation throughout the year.

Phase Shift

The phase shift of a trigonometric function indicates how the function is shifted horizontally along the time axis. For the function \( T(t) = 17 \cos\left(\frac{\pi}{6}t - \frac{7\pi}{6}\right) + 75 \), the phase shift results from the expression \( -\frac{7\pi}{6} \).
To compute the phase shift, solve \( \frac{\pi}{6}t - \frac{7\pi}{6} = 0 \) to find \( t \). This gives us \( t = 7 \), meaning the cosine wave starts at its baseline position after being shifted 7 months to the right. Understanding phase shift is crucial in determining the timing of temperature changes throughout the year.

Vertical Shift

The vertical shift in a trigonometric function is the constant added or subtracted from the main trigonometric expression, which, in this function, is expressed as \( +75 \). This shift moves the entire wave up or down along the temperature axis.
For \( T(t) = 17 \cos\left(\frac{\pi}{6}t - \frac{7\pi}{6}\right) + 75 \), the vertical shift is 75, which means that the entire wave is moved up by 75 degrees Fahrenheit. This new position reflects an average (or baseline) temperature around which the annual fluctuations occur. Vertical shifts help identify the average condition or baseline measurement in periodic phenomena.

Cosine Function

The cosine function is a periodic function characterized by its wave-like appearance. It repeats every cycle with a consistent shape. In our function example, \( T(t) = 17 \cos\left(\frac{\pi}{6}t - \frac{7\pi}{6}\right) + 75 \), the core mathematical term is the cosine expression.
The cosine function helps model periodic events effectively, such as seasonal temperature changes, because of its rise and fall around a central axis over regular intervals. Cosine functions are widely used in physics, engineering, and other fields to study cyclical data and predict future trends based on past behaviors.

Temperature Modeling

Temperature modeling with trigonometric functions aims to simulate and predict temperature behavior over time. By using functions like \( T(t) = 17 \cos\left(\frac{\pi}{6}t - \frac{7\pi}{6}\right) + 75 \), we can approximate real-world temperature patterns with mathematical expressions.
This approach uses features such as amplitude, phase shifts, and vertical shifts to reflect the natural highs and lows in temperature, providing useful insights for planning in agriculture, energy management, and infrastructure design. Temperature models help us prepare for climatic changes by understanding past temperature trends and anticipating future conditions.