a fungus has a height of 10cm. it grows by 3% every 30 days. what shall be its height after 120 days?

To find the height of the fungus after 120 days, given that it grows by 3% every 30 days, we can use the formula for exponential growth:

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

where:
- \(H(t)\) is the height of the fungus after \(t\) days.
- \(H_0\) is the initial height of the fungus.
- \(r\) is the growth rate per period.
- \(n\) is the number of growth periods.

### Given Values:
- Initial height, \(H_0 = 10 \text{ cm}\)
- Growth rate, \(r = 3\% = 0.03\)
- Growth period = 30 days
- Total time, \(t = 120 \text{ days}\)

### Step-by-Step Calculation:

1. **Calculate the number of growth periods (\(n\)) in 120 days**:
   \[
   n = \frac{t}{\text{growth period}} = \frac{120}{30} = 4
   \]

2. **Apply the growth formula**:
   \[
   H(120) = 10 \times (1 + 0.03)^4
   \]

3. **Compute the result**:
   \[
   H(120) = 10 \times (1.03)^4
   \]

Calculating \((1.03)^4\):

\[
(1.03)^4 \approx 1.1255
\]

Therefore:

\[
H(120) = 10 \times 1.1255 = 11.255 \text{ cm}
\]

### Conclusion:

The height of the fungus after 120 days will be approximately **11.255 cm**.

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Would you like to see more details on the calculation, or do you have any questions?

### Related Questions:
1. What would be the height of the fungus after 150 days?
2. How long would it take for the fungus to double its height?
3. If the growth rate were 5% every 30 days, what would be the height after 120 days?
4. What is the continuous growth rate equivalent to a 3% growth every 30 days?
5. How does changing the initial height affect the height after 120 days?

### Tip:
For exponential growth, small changes in the growth rate or the number of periods can have a significant impact on the final result.

Explore how to calculate the height of a fungus growing exponentially by 3% every 30 days. Use our step-by-step guide and formula to determine the fungus's height after 120 days. Suitable for Grades 9-12.

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