assume the substance has a half-life of 11 years and the initial amount is 126 grams.How long will it be until only 15 % remains?
To determine how long it will take for only 15% of the substance to remain, we need to use the concept of exponential decay.
First, let's define the equation for exponential decay:
A = A₀ * (1/2)^(t/h)
Where:
A₀ is the initial amount of the substance,
A is the remaining amount of the substance after time t,
h is the half-life of the substance,
t is the time elapsed.
In this case, we're given that the initial amount is 126 grams (A₀ = 126 grams), and we want to find the time (t) required for 15% (0.15) of the substance to remain (A = 0.15 * A₀).
Plug in the values into the equation:
0.15 * A₀ = A₀ * (1/2)^(t/11)
Now, we can cancel out A₀ on both sides:
0.15 = (1/2)^(t/11)
Taking the logarithm of both sides (base 2 since we have 1/2):
log₂(0.15) = log₂((1/2)^(t/11))
Using the logarithmic property (logₐ(b^c) = c * logₐ(b)):
log₂(0.15) = (t/11) * log₂(1/2)
Now, solve for t by isolating it:
t/11 = log₂(0.15) / log₂(1/2)
Simplify the right side using the property log₂(b) = log(c)/log(b):
t/11 = log(0.15) / log(1/2)
Finally, solve for t:
t = 11 * (log(0.15) / log(1/2))
Using a calculator, we find that t ≈ 30.69 years.
Therefore, it will take approximately 30.69 years for only 15% of the substance to remain.
To determine how long it will take for only 15% of the substance to remain, we need to calculate the number of half-lives it will take for the amount to decrease to 15% of the initial amount.
Step 1: Calculate the amount of the substance that remains after each half-life.
After the first half-life: Remaining amount = Initial amount * (1/2) = 126 grams * (1/2) = 63 grams
After the second half-life: Remaining amount = Previous remaining amount * (1/2) = 63 grams * (1/2) = 31.5 grams
After the third half-life: Remaining amount = Previous remaining amount * (1/2) = 31.5 grams * (1/2) = 15.75 grams
Step 2: Calculate the percentage of the initial amount that remains after the number of half-lives.
Percentage remaining = (Remaining amount / Initial amount) * 100
After three half-lives: Percentage remaining = (15.75 grams / 126 grams) * 100 = 12.5%
Step 3: Determine the number of additional half-lives needed to reach 15% remaining.
Let's assume x is the number of additional half-lives needed.
Percentage remaining after x additional half-lives = (15/100) * 126 grams
Setting up the equation:
(1/2)^x * 126 grams = (15/100) * 126 grams
Dividing both sides by 126 grams:
(1/2)^x = 15/100
Taking the logarithm (base 2) of both sides:
log2((1/2)^x) = log2(15/100)
Using the logarithmic identity: log_b(a^c) = c * log_b(a)
x * log2(1/2) = log2(15/100)
Since log2(1/2) is equal to -1:
-x = log2(15/100)
Dividing both sides by -1:
x = -log2(15/100)
Using a calculator:
x ≈ 3.91 (rounded to two decimal places)
Therefore, it will take approximately 3.91 additional half-lives for only 15% of the initial amount (126 grams) to remain.
1/2 = e^k(11)
ln .5 = 11 k
11 k = -.5978
k = -.05435
so
.15 = e^-.05435 t
-1.897 = - .05435 t
t = 34.9 years