Problem 6 - College Algebra

Question

The temperature of an object in degrees Fahrenheit after \(t\) minutes is represented by the equation \(T(t)=68 e^{-0.0174 t}+72 .\) To the nearest degree, what is the temperature of the object after one and a half hours?

Step-by-Step Solution

Answer

The temperature is 86°F after 1.5 hours.

Step 1: Convert Hours to Minutes

The problem gives time in hours, but the formula uses minutes. We need to convert 1.5 hours to minutes. Since 1 hour has 60 minutes, we calculate: 1.5 hours × 60 minutes/hour = 90 minutes.

Step 2: Substitute Time into the Equation

Substitute the converted time (90 minutes) into the equation for the temperature: \[ T(t) = 68 e^{-0.0174 \times 90} + 72 \]

Step 3: Calculate the Exponential Term

First, calculate the exponent: \[ -0.0174 \times 90 = -1.566 \] Then compute \( e^{-1.566} \) using a calculator: \[ e^{-1.566} \approx 0.209 \]

Step 4: Evaluate the Expression

Replace the exponential value back into the equation to solve for \( T(90) \): \[ T(90) = 68 \times 0.209 + 72 \] Calculate: \[ 68 \times 0.209 \approx 14.212 \] Then, add 72 to this result.

Step 5: Calculate the Final Temperature

Add the calculated multiplication to 72: \[ T(90) = 14.212 + 72 \approx 86.212 \]

Step 6: Round to the Nearest Degree

Finally, round 86.212 to the nearest whole number. The temperature of the object after one and a half hours, to the nearest degree, is 86°F.

Key Concepts

Temperature Conversion

Often, you'll need to convert time or temperature units when working with equations involving exponential functions. In our context, we started with time given in hours and needed to convert it to minutes to match the exponential function's requirements. This is a common necessity in scientific equations, where time intervals must align with other units in the equation to ensure coherence and accuracy.

To convert hours to minutes, simply multiply the number of hours by 60, since 1 hour equals 60 minutes. For our problem:

• 1.5 hours × 60 minutes/hour = 90 minutes

Temperature conversions, on the other hand, involve changing temperatures from one scale to another (e.g., Celsius to Fahrenheit). Although our equation does not require changing temperature scales, it's useful to know the general formula for common conversions, such as:

• Fahrenheit to Celsius: \( C = \frac{5}{9}(F - 32) \)
• Celsius to Fahrenheit: \( F = \frac{9}{5}C + 32 \)

Exponential Decay

Exponential decay describes processes where quantities decrease over time at a rate proportional to their current value. This can be used in various scenarios such as population decline, radioactive decay, or cooling objects, as in our case. The equation provided, \( T(t) = 68 e^{-0.0174 t} + 72 \), expresses such a decay in context: cooling of an object over time.

Here, \( e^{-0.0174 t} \) is the exponential decay factor. This part of the function goes to zero as \( t \) increases, indicating that the temperature approaches a steady value (72 degrees in this case) over time. This is typical behavior in exponential decay functions where the function dwindles towards an asymptotic value.

In solving this problem:

• We computed the exponential term at a specific instant (90 minutes) to see the rate and extent of decay.
• Thus, we found \( e^{-1.566} \approx 0.209 \), signifying a significant decay from the initial state.

Rounding Numbers

Rounding numbers simplifies results so they are easier to understand or use, especially when precision beyond a certain point is not necessary. In our solution, after calculating the temperature at 90 minutes, we obtained a raw result of approximately 86.212 degrees.

Rounding to the nearest whole number requires examining the first decimal place:

• If this digit is 5 or more, you round up.
• If it's less than 5, you round down.

So, since the first decimal is 2 in 86.212, we round down to 86. This process of rounding not only makes the final result more digestible but also reflects an appropriate precision level for practical applications.

Rounding plays a vital role in reporting results succinctly while retaining meaningfulness and ensuring that the precision reported matches the least precise input data or expectations.