Absolutely, let’s walk through the steps to determine the value of the quantity after 21 days.
1. Convert Total Time to Weeks:
Given that the total time is 21 days, let's first convert this period into weeks. As there are 7 days in a week:
[tex]\[
\text{Total time in weeks} = \frac{21 \text{ days}}{7 \text{ days/week}} = 3 \text{ weeks}
\][/tex]
2. Determine the Number of Doublings:
The quantity doubles every 2 weeks. To find out how many times it doubles over the period of 3 weeks:
[tex]\[
\text{Number of doublings} = \frac{\text{Total time in weeks}}{\text{Doubling time in weeks}} = \frac{3 \text{ weeks}}{2 \text{ weeks}} = 1.5
\][/tex]
3. Calculate the Final Value of the Quantity:
The initial value of the quantity is 830. With the number of doublings calculated as 1.5, we use the formula for exponential growth:
[tex]\[
\text{Final value} = \text{Initial value} \times (2^{\text{Number of doublings}})
\][/tex]
Plugging in the values:
[tex]\[
\text{Final value} = 830 \times (2^{1.5}) = 830 \times \sqrt{2^3} = 830 \times \sqrt{8} \approx 830 \times 2.828
\][/tex]
[tex]\[
\text{Final value} \approx 2347.594513539338
\][/tex]
4. Round to the Nearest Hundredth:
Finally, to provide the value rounded to the nearest hundredth:
[tex]\[
\text{Rounded final value} \approx 2347.59
\][/tex]
To summarize, after 21 days, the value of the quantity is approximately 2347.59, rounded to the nearest hundredth.
