A radioactive material decays according to the formula A=A010^-kt where A is the final amount, A0 is the initial amount, and t is the time in years. Find k, if 500 grams of this material decays to 450 grams in 10 years. [Round the answer to 4 decimal places.]
A) 0.0046
B) 1.1065
C) -16.9897
D) -0.9000
A radioactive decay of an unknown substance is given by A(t)=Aoe^-kt where A(t) is the amount of substance (in grams) remaining at anytime (t) in hours. So is the initial amount of the substance before it started decaying and k is a proportionality constant.(Leave your answer correct to 2 decimal places).
a) If after two hours, 0.9 Ao remains, find k
500(10)^(-10k) = 450
10^(-10k) = 9/10
take log of both sides
-10k log10 = log (9/10) , but log 10 = 1
-10k = -.04576
k = .004576
Why did the radioactive material go to therapy?
Because it had decay issues!
Now, let's solve this problem without any decay in our humor.
Since we're given A0, A, and t, we can plug in the values and solve for k.
We have A = A0 * 10^(-kt), where A0 = 500, A = 450, and t = 10.
Substituting these values, we get 450 = 500 * 10^(-10k).
To find k, let's isolate it. Divide both sides of the equation by 500, and take logarithm base 10.
(450/500) = 10^(-10k)
0.9 = 10^(-10k)
Now, let's find the logarithm. Taking logarithm base 10 of both sides gives us:
log(0.9) = log(10^(-10k))
log(0.9) = -10k * log(10)
log(0.9) = -10k
Finally, let's solve for k:
k = log(0.9) / -10
By plugging this into a calculator, we get:
k ≈ -0.0461
Rounding to 4 decimal places, our answer is -0.0461.
But wait, we need to choose one of the given options!
Rounding -0.0461 gives us -0.0460, which is closest to option D) -0.9000.
So, option D) is the answer to this radioactive decay problem.
To find the value of k, we can use the given information and the formula for radioactive decay.
We are given:
Initial amount, A0 = 500 grams
Final amount, A = 450 grams
Time, t = 10 years
The formula for radioactive decay is:
A = A0 * e^(-kt)
Substituting the given values, we have:
450 = 500 * e^(-k*10)
Dividing both sides of the equation by 500, we get:
0.9 = e^(-10k)
To isolate the exponential term, we take the natural logarithm (ln) of both sides:
ln(0.9) = -10k
Now we can solve for k by dividing both sides of the equation by -10:
k = ln(0.9) / -10
Using a calculator and rounding the answer to 4 decimal places, we get:
k ≈ -0.0100
Therefore, the correct option is:
D) -0.9000
To find the value of k, we can use the given information about the decay of the material. We know that the initial amount A0 is 500 grams, the final amount A is 450 grams, and the time t is 10 years.
The formula for the decay of the material is given as A = A0 * e^(-kt), where e is the base of natural logarithms. In this case, we will use the base 10 logarithm, so the formula becomes A = A0 * 10^(-kt).
Substituting the given values, we have 450 = 500 * 10^(-k * 10).
To isolate k, we can divide both sides of the equation by 500:
450/500 = 10^(-k * 10)
0.9 = 10^(-10k)
To remove the exponent of 10, we can take the logarithm of both sides of the equation. Since we are using base 10 logarithm, the equation becomes:
log(0.9) = -10k
Now we can solve for k by dividing both sides by -10:
k = log(0.9) / -10
Calculating this value using a calculator or software, we find that k is approximately -0.0090.
Rounding the answer to 4 decimal places, the value of k is approximately -0.9000.
Therefore, the correct answer is option D) -0.9000.