Problem 6 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The temperature of the object after 90 minutes is 86°F.

Step 1: Convert Time to Minutes

The time provided is in hours, and we need to convert it to minutes. Since there are 60 minutes in an hour, one and a half hours is equal to \(1.5 \times 60 = 90\) minutes.

Step 2: Substitute Time into the Equation

Using the equation \(T(t)=68 e^{-0.0174 t}+72\), substitute \(t = 90\) into the equation to find the temperature: \[ T(90) = 68 e^{-0.0174 \times 90} + 72 \].

Step 3: Calculate the Exponential Part

First calculate the exponent: \(-0.0174 \times 90 = -1.566\). Then compute \(e^{-1.566}\). Using a calculator, \(e^{-1.566} \approx 0.2096\).

Step 4: Compute the Temperature

Now substitute \(e^{-1.566} \approx 0.2096\) back into the equation: \[ T(90) = 68 \times 0.2096 + 72 \]. Calculate this to find the temperature: \(68 \times 0.2096 = 14.2528\). Add this to 72 to get \(14.2528 + 72 \approx 86.2528\).

Step 5: Round the Temperature

Round the result \(86.2528\) to the nearest whole number. This gives us a temperature of \(86\) degrees Fahrenheit.

Key Concepts

Temperature Conversion

Temperature conversion is the process of changing the temperature reading from one unit to another. This is often needed in science and everyday life.
In our problem, we're dealing with time conversion, as the temperature equation requires time input in minutes. Here's what you need to know:

• 1 hour equals 60 minutes.
• To find the total minutes for any given hours, multiply the hours by 60.

For our exercise, 1.5 hours was converted to 90 minutes since 1.5 times 60 yields 90. This conversion is crucial because it aligns the units with the equation to determine the temperature effectively.

Exponential Function

Exponential functions are a type of mathematical function that involve the constant e (approximately 2.71828), which is the base of the natural logarithm. These functions have the general form \(f(t) = a e^{bt} + c\), where:

• \(a\) is the initial amount.
• \(b\) is the growth (or decay) rate.
• \(t\) is time.
• \(c\) is a constant that shifts the function up or down.

In the equation given in the exercise, \(T(t) = 68e^{-0.0174t} + 72\), we see an exponential decay because the exponent \(-0.0174\) is negative. This means over time, the effect of the term with the exponential factor decreases.
When \(t = 90\) minutes is plugged into the function, it shows how the temperature changes at that point, reflecting the cooling effect described by the exponential decay.

Rounding Numbers

Rounding numbers is a mathematical technique used to approximate a number to the nearest desired decimal or whole number. This is important for simplifying data and ensuring results are understandable and manageable. Here's a simple way to round:

• If the digit to the right of your rounding place is 5 or greater, round up.
• If it is less than 5, round down.

In our example, after calculating the temperature using the equation, we arrived at 86.2528. To provide a clear, whole number result, we rounded this value.
We look at the digit after the decimal point (2), which is less than 5, so we round down the temperature to 86 degrees Fahrenheit.