Approximately 3.5355 mg of the sample will remain after 4000 years.
To determine how much of the sample will remain after 4000 years.
We can use the formula for exponential decay:
N(t) = N₀ * (1/2)^(t / T)
Where:
N(t) is the amount remaining after time t
N₀ is the initial amount
T is the half-life
Given:
Initial amount (N₀) = 20 mg
Half-life (T) = 1600 years
Time (t) = 4000 years
Plugging in the values, we get:
N(4000) = 20 * (1/2)^(4000 / 1600)
Simplifying the equation:
N(4000) = 20 * (1/2)^2.5
N(4000) = 20 * (1/2)^(5/2)
Using the fact that (1/2)^(5/2) is the square root of (1/2)^5, we have:
N(4000) = 20 * √(1/2)^5
N(4000) = 20 * √(1/32)
N(4000) = 20 * 0.1767766953
N(4000) ≈ 3.5355 mg
Therefore, approximately 3.5355 mg of the sample will remain after 4000 years.
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Jeff has 32,400 pairs of sunglasses. He wants to distribute them evenly among X people, where X is
a positive integer between 10 and 180, inclusive. For how many X is this possible?
Answer:
To distribute 32,400 pairs of sunglasses evenly among X people, we need to find the positive integer values of X that divide 32,400 without any remainder.
To determine the values of X for which this is possible, we can iterate through the positive integers from 10 to 180 and check if 32,400 is divisible by each integer.
Let's calculate:
Number of possible values for X = 0
For each value of X from 10 to 180, we check if 32,400 is divisible by X using the modulo operator (%):
for X = 10:
32,400 % 10 = 0 (divisible)
for X = 11:
32,400 % 11 = 9 (not divisible)
for X = 12:
32,400 % 12 = 0 (divisible)
...
for X = 180:
32,400 % 180 = 0 (divisible)
We continue this process for all values of X from 10 to 180. If the remainder is 0, it means that 32,400 is divisible by X.
In this case, the number of possible values for X is the count of the integers from 10 to 180 where 32,400 is divisible without a remainder.
After performing the calculations, we find that 32,400 is divisible by the following values of X: 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 90, 96, 100, 108, 120, 128, 135, 144, 150, 160, 180.
Therefore, there are 33 possible values for X between 10 and 180 (inclusive) for which it is possible to distribute 32,400 pairs of sunglasses evenly.
Hope it helps!