A quantity with an initial value of 400 grows exponentially at a rate such that the quantity doubles every 8 days. What is the value of the quantity after 29 days, to the nearest hundredth?
To solve this problem, we can use the formula for exponential growth:
\[Q(t) = Q_0 \times 2^{\left(\frac{t}{d}\right)}\]
Where:
- \(Q(t)\) is the quantity after time \(t\),
- \(Q_0\) is the initial quantity,
- \(d\) is the doubling time period,
- \(t\) is the time period we are interested in.
Given:
- \(Q_0 = 400\),
- \(d = 8\),
- \(t = 29\).
Plugging in these values, we get:
\[Q(29) = 400 \times 2^{\left(\frac{29}{8}\right)}\]
Calculating the exponent:
\[Q(29) = 400 \times 2^{3.625}\]
\[Q(29) = 400 \times 16.6793\]
\[Q(29) ≈ 6671.72\]
Rounded to the nearest hundredth, the value of the quantity after 29 days is approximately 6671.72.