Explain how you would solve the following problem. For the first 12 months, a baby's height grows exponentially by about 7.3% each month. Your teacher’s baby, Michael, was 20 inches tall at birth. After how many months will Michael be 30 inches tall?

To solve this problem, we can model the baby's height growth using an **exponential growth formula**. The general formula for exponential growth is:

\[
H(t) = H_0 \times (1 + r)^t
\]

Where:
- \( H(t) \) is the height after \( t \) months,
- \( H_0 \) is the initial height (at \( t = 0 \)),
- \( r \) is the monthly growth rate (as a decimal),
- \( t \) is the time in months.

### Step-by-Step Solution:

1. **Identify the known values:**
   - Initial height \( H_0 = 20 \) inches (at birth),
   - Growth rate \( r = 7.3\% = 0.073 \),
   - Final height \( H(t) = 30 \) inches.

2. **Set up the equation:**

   We want to find \( t \), the number of months it takes for the baby's height to grow from 20 inches to 30 inches. Using the formula:

   \[
   30 = 20 \times (1 + 0.073)^t
   \]

   This simplifies to:

   \[
   30 = 20 \times (1.073)^t
   \]

3. **Isolate the exponential term:**

   Divide both sides by 20:

   \[
   1.5 = (1.073)^t
   \]

4. **Take the natural logarithm of both sides:**

   To solve for \( t \), we take the natural logarithm (ln) of both sides:

   \[
   \ln(1.5) = \ln((1.073)^t)
   \]

   Using the logarithmic property \( \ln(a^b) = b \cdot \ln(a) \), we get:

   \[
   \ln(1.5) = t \cdot \ln(1.073)
   \]

5. **Solve for \( t \):**

   Now, divide both sides by \( \ln(1.073) \):

   \[
   t = \frac{\ln(1.5)}{\ln(1.073)}
   \]

6. **Calculate the values:**

   Using a calculator to find the natural logarithms:

   \[
   \ln(1.5) \approx 0.4055 \quad \text{and} \quad \ln(1.073) \approx 0.0705
   \]

   So:

   \[
   t \approx \frac{0.4055}{0.0705} \approx 5.75
   \]

Thus, it will take approximately **5.75 months** (or about 5 months and 3 weeks) for Michael to reach 30 inches in height.

### Summary:
Michael will be about 30 inches tall after approximately **5.75 months**.

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:

1. What would be Michael's height after 8 months?
2. If the growth rate were different, say 5% per month, how would that change the number of months required to reach 30 inches?
3. What if Michael was 18 inches at birth? How would that change the solution?
4. How can logarithms help solve different kinds of exponential growth problems?
5. How would you solve the same problem if the growth rate decreased each month?

**Tip:** Logarithms are a powerful tool for solving exponential equations! Remember, whenever an unknown is in the exponent, logarithms help "bring it down" to a manageable level.

For the first 12 months, a baby's height grows exponentially by about 7.3% each month. Your teacher’s baby, Michael, was 20 inches tall at birth. After how many months will Michael be 30 inches tall?

Solve an exponential growth problem: Find out how many months it will take for a baby’s height to grow from 20 inches to 30 inches, given a 7.3% monthly growth rate. This problem uses logarithmic functions and exponential growth models, suitable for high school students in Grades 9-12.

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