There is mold growing in my fridge. The mole frozen to re-proportional to the amount of existing mold. At noon, there were 0.4 g mold in the fridge. This evening at 9 PM there are now 1.6 g of mold in the fridge. How many grams of mold were there at 5 AM this morning? Please transfer the 23 decimal places. Enter only the number.

To solve this, we first assume the mold grows exponentially since it is proportional to the amount of existing mold. The exponential growth model is generally given by:

\[
M(t) = M_0 \cdot e^{kt}
\]

Where:
- \(M(t)\) is the amount of mold at time \(t\),
- \(M_0\) is the initial amount of mold,
- \(k\) is the growth rate,
- \(t\) is the time.

We know:
- At noon (\(t = 0\)), \(M(0) = 0.4\) g.
- At 9 PM (\(t = 9\) hours), \(M(9) = 1.6\) g.

Let's calculate \(k\), the growth rate.

The equation at \(t = 9\) is:
\[
1.6 = 0.4 \cdot e^{9k}
\]

Solving for \(k\):

\[
4 = e^{9k}
\]
\[
9k = \ln(4)
\]
\[
k = \frac{\ln(4)}{9}
\]

Now, to find the amount of mold at 5 AM, which is 7 hours before noon (\(t = -7\)), we use the equation:

\[
M(-7) = 0.4 \cdot e^{-7k}
\]

Substituting the value of \(k\):

\[
M(-7) = 0.4 \cdot e^{-7 \cdot \frac{\ln(4)}{9}}
\]

Now, let’s calculate this.The amount of mold in the fridge at 5 AM was approximately 0.1360790000174377000000000 grams.

Do you have any questions or would you like further details?

Here are 5 related questions to explore:
1. How can we model other types of growth, such as logistic growth?
2. What happens if the mold growth rate decreases over time?
3. How does changing the initial amount of mold affect future growth?
4. Can we model decay (shrinking instead of growing) in the same way?
5. How is exponential growth used in real-world applications, such as population growth?

**Tip:** Exponential growth occurs when the rate of change is proportional to the current value. It's useful for modeling populations, bacteria growth, and radioactive decay.

Solve the exponential growth problem where mold in a fridge grows from 0.4 g to 1.6 g between noon and 9 PM. Learn how to calculate the initial amount of mold at 5 AM using the exponential growth formula and logarithms. Suitable for high school to undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation