A quantity with an initial value of 510 grows exponentially at a rate such that the quantity doubles every 8 decades. What is the value of the quantity after 7 years, to the nearest hundredth?
To solve this problem, we first need to calculate the number of doubling periods that have occurred in 7 years.
Since the quantity doubles every 8 decades, it will have doubled 7/8 = 0.875 times in 7 years.
To find the value of the quantity after 7 years, we can use the formula for exponential growth:
Q(t) = Q0 * 2^(t/d)
Where:
- Q(t) is the quantity after time t
- Q0 is the initial quantity
- t is the time that has passed
- d is the number of doubling periods that have occurred
Plugging in the values given:
Q(7) = 510 * 2^(0.875)
Q(7) = 510 * 2^(7/8)
Q(7) = 510 * 2^(7/8)
Q(7) = 510 * 2^(0.875)
Q(7) ≈ 510 * 2^0.875
Q(7) ≈ 510 * 2.3784
Q(7) ≈ 1213.884
Therefore, the value of the quantity after 7 years is approximately 1213.88 to the nearest hundredth.