Problem 9 - 5 Steps to A 5: AP Calculus AB 2021

Question

The change of temperature of a cup of coffee measured in degrees Fahrenheit in a certain room is represented by the function \(f(t)=-\cos \left(\frac{t}{4}\right)\) for \(0 \leq t \leq 5,\) where \(t\) is measured in minutes. If the temperature of the coffee is initially \(92^{\circ} \mathrm{F},\) find its temperature after the first 5 minutes.

Step-by-Step Solution

Answer

Answer: The temperature of the coffee after the first 5 minutes is approximately \(91.239^{\circ}\mathrm{F}\).

Step 1: Find the Change in Temperature After 5 Minutes

To find the change in temperature after 5 minutes, we will plug in \(t=5\) into the function \(f(t)\). $$ f(5)=-\cos\left(\frac{5}{4}\right) $$

Step 2: Evaluate the Function f(5)

Now evaluate the function at \(t=5\) $$ f(5)=-\cos\left(\frac{5}{4}\right)\approx -0.761 $$ The change in temperature after the first 5 minutes is approximately -0.761°F.

Step 3: Calculate the Final Temperature

To find the temperature of the coffee after 5 minutes, add the change in temperature to the initial temperature. $$ T_{final} = T_{initial} + f(5) $$ $$ T_{final} = 92°F - 0.761°F \approx 91.239°F $$ The temperature of the coffee after the first 5 minutes is approximately \(91.239^{\circ}\mathrm{F}\).

Key Concepts

Temperature Change

When solving problems that involve temperature change, we are often interested in how temperature varies over a specific period. In the context of the given exercise, the temperature change of a cup of coffee is described by a mathematical function that shows the temperature fluctuation over time.

The important factors in such problems usually include:

• The initial temperature (known at the start).
• The rate of temperature change, formulated as a function of time.
• The duration over which the temperature change is considered.

The exercise uses a trigonometric function, specifically a cosine function, to represent these changes over a given time span. These functions are great for modeling periodic or cyclic behaviors, which are common in real-world scenarios like cooling coffee or warming up.
Remember, the final step in such problems is summing up the initial temperature and the change calculated using the function. This gives us the new temperature after the given timeframe.

Function Evaluation

Function evaluation is the process of determining the output of a function for a specific input value. In our exercise, the function given is \(f(t) = -\cos\left(\frac{t}{4}\right)\) and we are tasked with evaluating it at \(t=5\).

The steps to evaluate a function like this one include:

• Substitute the given input \(t\) into the function.
• Perform the appropriate arithmetic operations including division and applying trigonometric calculations.
• Evaluate the trigonometric function value.

In this specific context, \(f(5) = -\cos\left(\frac{5}{4}\right)\), which implies substituting 5 into the function and calculating the cosine of \(\frac{5}{4}\) radians. Trigonometric values can often require calculators for precise results, as they might not correspond to simple fractions of \(\pi\).
The result \(f(5) \approx -0.761\) signifies how much the temperature of the coffee changes after a specified duration, in this case, 5 minutes.

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in mathematics for modeling cycles, waves, and oscillations. In our exercise, we use the cosine function to describe the change in temperature of coffee over time. This is because such functions are periodic and can effectively model fluctuations.

Important aspects of trigonometric functions include:

• Periodic nature: They repeat values in cycles.
• Range of values: Cosine, for instance, oscillates between -1 and 1.
• Use of radians: Often, trigonometric functions use angles measured in radians.

The function \(-\cos\left(\frac{t}{4}\right)\) introduces a negative cosine to reflect a decrease in temperature over time, simulating how coffee cools down. The multiplication of \(t\) by \(\frac{1}{4}\) effectively stretches the cosine function, altering the rate at which these oscillations occur.
Understanding trigonometric functions is crucial as they provide invaluable tools not only in mathematics but also in physics, engineering, signal processing, and various other fields requiring the modeling of cyclical behavior.