A quantity with an initial value of 670 decays continuously at a rate of 25% per hour. What is the value of the quantity after 7 hours, to the nearest hundredth?
To find the value of the quantity after 7 hours, we can use the formula for exponential decay:
\[Q(t) = Q(0)e^{kt}\]
Where:
- \(Q(t)\) is the quantity after time \(t\)
- \(Q(0)\) is the initial quantity
- \(k\) is the decay constant
- \(t\) is the time elapsed
In this case, we know that the initial quantity is 670 and the decay rate is 25% per hour. Since the decay rate is given as a percentage, we can write it as a decimal by dividing by 100: \(k = \frac{25}{100} = 0.25\).
Now we can substitute the values into the formula and solve for \(Q(7)\):
\[Q(7) = 670e^{0.25(7)}\]
Using a calculator, we find:
\[Q(7) \approx 670e^{0.25(7)} \approx 528.405\]
Rounded to the nearest hundredth, the value of the quantity after 7 hours is approximately 528.41.