A certain radioactive material is known to decay at a rate proportional to the
amount present. A block of this material originally having a mass of 100 grams is observed after 20 years to have a mass of only 80 grams. Find the half-life of this radioactive material. (Recall that the half-life is the length of time required for the material to be reduced by a half.)
A. 59.030 years
B. 61.045 years
C. 54.343 years
D. 56.442 years
E. 62.126 years
if the half-life is k years, then
y = 100(1/2)^(t/k)
since y(20) = 80,
100(1/2)^(20/k) = 80
k = 62.126
Plz dont cheat and use this website
Well, let me calculate the answer real quick, but I can't promise it won't be a "half-baked" response!
Given that the half-life is the amount of time required for the material to be reduced by half, we can use the formula:
final mass = initial mass * (1/2)^(t/half-life)
where t is the time passed and half-life is what we're trying to find.
Since the final mass is 80 grams and the initial mass is 100 grams, we can plug in the values:
80 = 100 * (1/2)^(20/half-life)
To find the half-life, we need to do a little math:
1/2^(20/half-life) = 80/100
1/2^(20/half-life) = 4/5
Now, let's make things simpler by taking the natural logarithm of both sides:
ln(1/2^(20/half-life)) = ln(4/5)
-20/half-life * ln(2) = ln(4/5)
Alright, now we can solve for the half-life:
-20 * ln(2) / half-life = ln(4/5)
ln(2) / half-life = ln(4/5) / -20
half-life = -ln(2) / (ln(4/5) / -20)
And calculating that value gives us... *drum roll please*...
B. 61.045 years!
So, the half-life of this radioactive material is approximately 61.045 years. Happy decaying!
To find the half-life of the radioactive material, we can use the exponential decay formula:
m(t) = m₀ * e^(-kt)
Where:
- m(t) is the mass of the material at time t,
- m₀ is the initial mass of the material,
- k is the decay constant.
We are given that after 20 years, the mass of the material is 80 grams. So we can write:
80 = 100 * e^(-20k)
Dividing both sides by 100:
0.8 = e^(-20k)
Taking the natural logarithm of both sides:
ln(0.8) = -20k
Now we can solve for k:
k = ln(0.8) / -20
Using a calculator:
k ≈ -0.03466
Now we can use the half-life formula, which is given by:
t₁/₂ = ln(2) / k
Substituting the value of k we found:
t₁/₂ = ln(2) / (-0.03466)
Using a calculator:
t₁/₂ ≈ 20.007 years
So the half-life of the radioactive material is approximately 20.007 years.
Therefore, the closest option to the correct answer is:
A. 59.030 years.
To find the half-life of the radioactive material, we can use the formula for exponential decay:
mass = initial mass * (1/2)^(time / half-life)
Let's substitute the given values into the formula:
80 grams = 100 grams * (1/2)^(20 years / half-life)
To isolate the half-life, we need to solve for it. Let's simplify the equation by dividing both sides by 100 grams:
0.8 = (1/2)^(20 years / half-life)
Now, take the logarithm (base 1/2) of both sides to eliminate the exponent:
log₂(0.8) = log₂((1/2)^(20 years / half-life))
Using the logarithmic identity logₐ(b^c) = c * logₐ(b), we can rewrite the equation as:
log₂(0.8) = (20 years / half-life) * log₂(1/2)
Since log₂(1/2) = -1, we have:
log₂(0.8) = -20 years / half-life
Now, let's solve for the half-life by isolating it:
half-life = -20 years / (log₂(0.8))
Using a calculator, we find that log₂(0.8) is approximately -0.3219. Substituting this value into the equation:
half-life ≈ -20 years / (-0.3219)
half-life ≈ 62.1087 years
Rounding to three decimal places, the half-life of the radioactive material is approximately 62.109 years.
Comparing this value to the given options, the closest option is:
E. 62.126 years
Therefore, the correct answer is E. 62.126 years.