Initially 700 milligrams of a radioactive substance was present. After 2 hours the mass had decreased by 5%. 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. (Round your answer to one decimal place.)
amount = 700 e^(kt)
when t = 2, amount = .95(700) = 665
665 = 700 e^(2k)
.95 = e^(2k)
take ln of both sides and use log rules:
lon .95 = 2k ln e
ln .95 = 2k
k = ln.95/2
amount = 700e^(ln.95/2 t) = 700 e^(-.025647t)
plug in t = 24 and find amount
let me know what you got, I wrote my answer down
A(2) = A0 e^(2k) = 0.95 A0
A(24) = A0 e^(24k) = A0 (e^2k)^12 = 0.95^12 A0
700*0.95^12 = _____
This doesn't give you the value of k, but it does give the same answer for A(24)
To construct an exponential model for the amount remaining of the decaying substance after t hours, we can use the formula:
A(t) = A0 * e^(kt)
Where:
A(t) is the amount remaining after t hours,
A0 is the initial amount,
k is the decay constant, and
t is the time in hours.
In this case, the initial amount A0 is 700 milligrams, and the mass has decreased by 5%, which means the remaining amount is 95% of the initial amount.
So we have A0 = 700 and A(t) = 0.95 * A0.
Substituting the values, we get:
0.95 * 700 = 700 * e^(k(2))
Simplifying further, we have:
665 = 700 * e^(2k)
Dividing both sides by 700, we get:
0.95 = e^(2k)
Now, to find the value of k, we can take the natural logarithm (ln) of both sides:
ln(0.95) = ln(e^(2k))
ln(0.95) = 2k
Dividing by 2:
k = ln(0.95) / 2
Using a calculator, we find:
k ≈ -0.0253
Now that we have the value of k, we can find the amount remaining after 24 hours by substituting t = 24 into the formula:
A(24) = 700 * e^(-0.0253 * 24)
Using a calculator, we get:
A(24) ≈ 370.4
Therefore, the amount remaining after 24 hours is approximately 370.4 milligrams.
To construct an exponential model for the amount remaining of the decaying substance after t hours, we can use the equation A(t) = A0e^kt.
In this case, A0 represents the initial amount of the substance, t represents the time in hours, and A(t) represents the amount remaining after t hours.
In the given problem, the initial amount A0 is 700 milligrams and the mass had decreased by 5%. A decrease of 5% means that 95% of the substance remains after 2 hours.
To find the value of k, we can use the percentage decay formula, which states that:
A(t) = A0 * (1 - r)^t
In this case, r is the decay rate, given by 5% or 0.05.
So, we have:
0.95A0 = A0 * (1 - 0.05)^2
Simplifying:
0.95 = (0.95)^2
Taking the square root of both sides:
√0.95 = 0.95
0.975 = 0.95
Now, we can find the value of k using the equation:
k = ln(0.975) / 2
Using a calculator, we find:
k ≈ -0.0122
Now that we have the value of k, we can substitute it into the exponential model equation to find the amount remaining after 24 hours:
A(24) = 700e^(-0.0122*24)
Calculating this expression:
A(24) ≈ 700e^(-0.2928)
Using a calculator, we find:
A(24) ≈ 314.3
Therefore, the amount remaining after 24 hours is approximately 314.3 milligrams.