https://file.aiquickdraw.com/mathbot/file/1728685883915no6phcc1.jpg

The problem provides the following temperature function for a bottle of water placed inside a cooler:

\[
C(t) = 8 + 12e^{-0.028t}
\]

where:
- \( C(t) \) is the temperature of the water at time \( t \) (in minutes),
- \( e \) is Euler's number (approximately 2.718),
- Deon wants the water to cool down to 16°C.

We are tasked with finding how long \( t \) (in minutes) it will take for the water to reach 16°C.

### Step-by-Step Solution:

1. Set \( C(t) = 16 \) and solve for \( t \):

\[
16 = 8 + 12e^{-0.028t}
\]

2. Subtract 8 from both sides:

\[
16 - 8 = 12e^{-0.028t}
\]

\[
8 = 12e^{-0.028t}
\]

3. Divide both sides by 12:

\[
\frac{8}{12} = e^{-0.028t}
\]

\[
\frac{2}{3} = e^{-0.028t}
\]

4. Take the natural logarithm (ln) of both sides to eliminate the exponential:

\[
\ln\left(\frac{2}{3}\right) = -0.028t
\]

5. Solve for \( t \):

\[
t = \frac{\ln\left(\frac{2}{3}\right)}{-0.028}
\]

Using a calculator:

\[
\ln\left(\frac{2}{3}\right) \approx -0.4055
\]

\[
t \approx \frac{-0.4055}{-0.028} \approx 14.5 \, \text{minutes}
\]

### Answer:
Deon should leave the bottle in the cooler for approximately **14.5 minutes**.

---

Would you like more details or have any other questions?

Here are some related questions:
1. How would the result change if the cooling constant were larger (say, \(-0.035\))?
2. How does the value of Euler's number \( e \) affect exponential cooling?
3. What does the time constant (\( -0.028 \)) represent in this cooling process?
4. How could you use this formula to estimate the water's temperature after 30 minutes?
5. What happens to the function \( C(t) \) as \( t \to \infty \)?

**Tip:** Always check units and variables carefully when working with real-life functions to ensure accuracy in modeling.

Deon places a bottle of water inside a cooler. As the water cools, its temperature C(t) in degrees Celsius is given by the function C(t) = 8 + 12e^{-0.028t}, where t is the number of minutes since the bottle was placed in the cooler. Deon wants to drink the water when it reaches a temperature of 16 degrees Celsius. How many minutes should he leave it in the cooler?

Learn how to solve an exponential cooling problem where the temperature of water placed in a cooler is modeled by the equation C(t) = 8 + 12e^{-0.028t}. This problem involves using natural logarithms to determine how long it takes for the water to reach 16°C. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation