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Chapter 7: Problem 110

The following function approximates the average monthly temperature \(y\) (in \(^{\circ} \mathrm{F}\) ) in Phoenix, Arizona. Here \(x\) represents the month, where \(x=1\) corresponds to January, \(x=2\) corresponds to February, and so on. $$ f(x)=19.5 \cos \left[\frac{\pi}{6}(x-7)\right]+70.5 $$ When is the average monthly temperature (a) \(70.5^{\circ} \mathrm{F}\) (b) \(55^{\circ} \mathrm{F} ?\)

Short Answer

Expert verified
(a) The average monthly temperature is \(70.5^{\u05df}F^{\text{\text{}F}\) in April (x=4) and October (x=10). (b) The average monthly temperature is \(55^{\text{}F}\) in mid-April (x= 4.75) and mid-November (x= 11.25).

Step by step solution

01

- Understanding the function

The given function approximates the average monthly temperature, where the month is represented by the variable \(x\) and the temperature by \(y = f(x)\). The function is \(f(x)=19.5 \, \text{cos} \left[\frac{\pi}{6}(x-7)\right]+70.5\).
02

- Solving for 70.5掳F

Set \(f(x)=70.5\) and solve for \(x$$70.5 = 19.5 \, \text{cos} \left[\frac{\pi}{6}(x-7)\right]+70.5\)Subtract 70.5 from both sides:\(0 = 19.5 \, \text{cos} \left[\frac{\pi}{6}(x-7)\right]\)Divide by 19.5:\(0 = \text{cos} \left[\frac{\pi}{6}(x-7)\right]\)Since the cosine function equals zero at \(\frac{\pi}{2} + k\pi\) for \(k \in \mathbb{Z}\), set:\(\frac{\pi}{6}(x-7) = \frac{\pi}{2} + k\pi\)Multiply by 6: \(\pi(x-7) = 3\pi + 6k\pi\)Simplify and solve for \(x\):\(x-7 = 3 + 6k$$x = 10 + 6k\). Since \(x\) must be a month (1 through 12),\(k = -1 \rightarrow x = 4\) (April)\(k = 0 \rightarrow x = 10\) (October)
03

- Solving for 55掳F

Set \(f(x)=55\) and solve for \(x$$55 = 19.5 \, \text{cos} \left[\frac{\pi}{6}(x-7)\right]+70.5\)Subtract 70.5 from both sides:\(-15.5 = 19.5 \, \text{cos} \left[\frac{\pi}{6}(x-7)\right]\)Divide by 19.5:\(\text{cos} \left[\frac{\pi}{6}(x-7)\right] = -\frac{15.5}{19.5}\)Calculate \(-\frac{15.5}{19.5} \approx -0.7949\), Let \(A = \text{cos}^{-1}(-0.7949)\approx 2.4871\),Then \(\frac{\pi}{6}(x-7) = 2.4871 + 2k\pi\) (cosine function period \(2\pi\))\(\frac{\pi}{6}(x-7) = 2.4871\) or \(\frac{\pi}{6}(x-7) = -2.4871 + 2\pi = 3.7961\)Solve each equation:1. \(\pi(x-7) = 14.9226\rightarrow x = 4.75\) (April)2. \(\pi(x-7) = 22.7766\rightarrow x=11.25\) (November)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

trigonometric functions
Trigonometric functions are essential in understanding periodic phenomena, such as the average monthly temperature in Phoenix, Arizona. The given function approximates the temperature using a cosine function, which is one of the primary trigonometric functions. In our example, the function is represented as:
\[ f(x)=19.5 \, \cos \left[\frac{\pi}{6}(x-7)\right]+70.5 \]
The cosine function, like other trigonometric functions, includes parameters that affect its amplitude, period, phase shift, and vertical shift.
  • The amplitude (19.5) influences the height of the wave.
  • The period, determined by the fraction \frac{\pi}{6} modifies the frequency of the changes in temperature.
  • The phase shift (7) shifts the wave horizontally, aligning it with months of the year.
  • The vertical shift (70.5) moves the entire function up or down, centering the wave around a different vertical position.
solving trigonometric equations
Solving trigonometric equations involves finding all possible values of the variable that make the equation true. Let's consider the scenario where we want to find out when the average monthly temperature is 70.5掳F. We set up the equation:
\[ 70.5 = 19.5 \, \cos \left[\frac{\pi}{6}(x-7)\right] + 70.5 \]
Subtracting 70.5 from both sides simplifies to:
  • \ 0 = 19.5 \, \cos \left[\frac{\pi}{6}(x-7)\right] \
  • \ \cos \left[\frac{\pi}{6}(x-7)\right] = 0 \

The subsequent steps involve using known values where the cosine function equals zero, \[ \frac{\pi}{2} + k\pi \], and solving for the variable x with appropriate transformations. This results in finding that the average monthly temperature is 70.5掳F in April (x=4) and October (x=10).
inverse cosine function
Understanding the inverse cosine function (also known as arccosine) is crucial when solving for specific temperature values. For instance, to determine when the average temperature is 55掳F, we start with:
\[ 55 = 19.5 \, \cos \left[\frac{\pi}{6}(x-7)\right] + 70.5 \]
Which simplifies to:
\[ \cos \left[\frac{\pi}{6}(x-7)\right] = -\frac{15.5}{19.5} \]
Here, the inverse cosine function helps us find the angle whose cosine is \-0.7949\, leading to: \[ \cos^{-1}(-0.7949) \approx 2.4871 \]
Using this, we determine:
  • \ \frac{\pi}{6}(x-7) = 2.4871 + 2k\pi \
  • \ \frac{\pi}{6}(x-7) = -2.4871 + 2\pi = 3.7961 \

Solving these equations allows us to find the months when the temperature is approximately 55掳F, which are April (x=4.75) and November (x=11.25).
periodicity of trigonometric functions
The periodicity of trigonometric functions means they repeat their values in regular intervals. For the cosine function, this interval is \ 2\pi \ which represents one full cycle.
In our temperature example, interpreting the periodic behavior of the cosine function helps us understand that specific temperatures recur in a predictable manner. This is why the cosine function is suitable for modeling seasonal changes.
For instance, setting the temperature to 70.5掳F results in periodic solutions:
\[ x = 10 + 6k \]
Here, the solutions repeat every 6 months (one period of the function), illustrating how temperatures cycle through the year in a predictable manner. This periodic nature allows us to identify months corresponding to specific average temperatures efficiently, making trigonometric functions powerful tools in such applications.

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Most popular questions from this chapter

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