A quantity with an initial value of 830 grows exponentially at a rate such that the quantity doubles every 2 weeks. What is the value of the quantity after 21 days, to the nearest hundredth?


We are given that a quantity \( Q \) has an initial value of 830 and doubles every 2 weeks (14 days). We need to determine the value of \( Q \) after 21 days, using the exponential growth model.

The general formula for exponential growth is:

\[
Q(t) = Q_0 \cdot 2^{t/T}
\]

Where:
- \( Q(t) \) is the quantity after time \( t \),
- \( Q_0 \) is the initial quantity (830),
- \( T \) is the doubling period (14 days),
- \( t \) is the time elapsed (21 days).

### Step 1: Plug in the known values
\[
Q(21) = 830 \cdot 2^{21/14}
\]

### Step 2: Simplify the exponent
\[
Q(21) = 830 \cdot 2^{1.5}
\]

### Step 3: Calculate \( 2^{1.5} \)
\[
2^{1.5} = \sqrt{2^3} = \sqrt{8} \approx 2.828
\]

### Step 4: Calculate the final quantity
\[
Q(21) = 830 \cdot 2.828 \approx 2347.24
\]

Thus, the value of the quantity after 21 days is approximately **2347.24**.

Would you like more details or have any other questions?

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Here are 5 related questions to further your understanding:
1. How would the formula change if the doubling period were 3 weeks instead of 2?
2. How does exponential growth differ from linear growth?
3. What is the rate of growth per day for this particular problem?
4. How long will it take for the quantity to quadruple?
5. What would happen to the value after 35 days?

**Tip**: For exponential growth problems, doubling time is crucial; it defines how quickly the quantity increases.

A quantity with an initial value of 830 grows exponentially at a rate such that the quantity doubles every 2 weeks. What is the value of the quantity after 21 days, to the nearest hundredth?

Learn how to calculate the exponential growth of a quantity that doubles every 2 weeks. Starting with an initial value of 830, this problem demonstrates the use of the exponential growth formula to find the value after 21 days. Suitable for students in Grades 9-12.

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