Initially 600 milligrams of a radioactive substance was present. After 7 hours the mass had decreased by 3%. Construct an exponential model
A(t) = A0ekt for the amount remaining of the decaying substance after t hours. Find the amount remaining after 24 hours.
To construct an exponential model for the amount remaining of the decaying substance, we can use the formula:
A(t) = A0e^(kt)
Where:
- A(t) represents the amount remaining after t hours.
- A0 represents the initial amount (600 milligrams in this case).
- k represents the decay constant.
- t represents the time in hours.
Since we know that the mass decreases by 3% every 7 hours, we can find the decay constant, k, using the formula:
A(t) = A0(1 - r)^n
Where:
- r represents the decay rate (3% = 0.03 in decimal form).
- n represents the number of intervals (7 hours in this case).
Substituting the given values:
A(7) = 600(1 - 0.03)^1
A(7) = 600(0.97)
A(7) = 582 milligrams (approximately)
Now, we can solve for k by substituting the values into the exponential model:
582 = 600e^(7k)
To isolate e^(7k), divide both sides by 600:
582/600 = e^(7k)
0.97 = e^(7k)
Take the natural logarithm (ln) of both sides:
ln(0.97) = 7k
Now, divide both sides by 7 to solve for k:
k = ln(0.97)/7
k ≈ -0.00406
Now that we have the value of k, we can use the exponential model to find the amount remaining after 24 hours:
A(24) = 600e^(-0.00406*24)
A(24) ≈ 600e^(-0.09744)
A(24) ≈ 600(0.905216)
A(24) ≈ 543.13 milligrams (approximately)
Therefore, the amount remaining after 24 hours is approximately 543.13 milligrams.
To construct an exponential model, we can start by understanding the given information.
We are told that initially, there were 600 milligrams of the radioactive substance.
After 7 hours, the mass had decreased by 3%. This means that only 97% of the initial mass remains.
So, we can set up the equation A(t) = A0ekt, where A(t) represents the amount of substance remaining after t hours, A0 is the initial amount (600 mg), k is the decay constant, and t is the time in hours.
Since we know that after 7 hours, only 97% of the initial mass remains, we can substitute these values into the equation:
0.97A0 = A0e(7k)
Simplifying the equation, we get:
0.97 = e(7k)
To find the decay constant k, we can take the natural logarithm (ln) of both sides:
ln(0.97) = 7k
Now, we can solve for k:
k = ln(0.97)/7
Using a calculator, we find that k is approximately -0.0030197.
Finally, we can substitute the value of k into the equation A(t) = A0ekt and find the amount remaining after 24 hours (t = 24):
A(24) = 600e(-0.0030197 * 24)
Calculating this expression, we find that the amount remaining after 24 hours is approximately 520.45 milligrams.
so
600(.97) = 600 e^(7k)
.97 = e^(7k)
ln .97 = 7k lne
7k = ln .97
k = ln .97/7 = -.004351315
amount = 600 e^-.0043511315t
so when t = 24
amount = 600 e^-.104431..
= .900836(600) ----> 90.08% remain
= 540.50 mg