A 50 gram sample of radioactive strontium will decay to 32 grams in 18 years.
a. Find an equation that will model how much of our sample M will remain after t years.
(50-32)grams/18yrs = 1 Gram/yr.
Mr = M-1*T = M-T = 50 - 18 = 32 grams. = Mass remaining.
Correction: The decay rate is exponential.
50/e^x = 32, e^x = 50/32 = 1.5625, X = 0.446 = kt, K = 0.446/t = 0.446/18 = 0.0248.
Mr = M/e^kt = M/e^(0.0248t) = Mass remaining.
To find an equation that models how much of the sample remains after a certain number of years, we can use the exponential decay formula:
M = M₀ * e^(-kt)
Where:
- M is the mass of the radioactive substance at time t.
- M₀ is the initial mass of the radioactive substance.
- k is the decay constant.
- t is the time in years.
- e is the base of the natural logarithm, approximately 2.71828.
In this case:
- M₀ = 50 grams (initial mass)
- M = 32 grams (mass after 18 years)
- t = 18 years
We need to find the value of k to complete the equation. We can use the given information to find the decay constant.
The decay constant (k) can be calculated using the half-life formula:
t₁/₂ = ln(2) / k
Where:
- t₁/₂ is the half-life of the radioactive substance.
Given that the sample decays from 50 grams to 32 grams in 18 years, the difference in mass is 50 grams - 32 grams = 18 grams. The half-life is the time it takes for the original mass to reduce by half, so we have:
t₁/₂ = 18 years
Plugging this value into the half-life formula, we get:
18 = ln(2) / k
To solve for k, we rearrange the equation:
k = ln(2) / 18
Now that we have the decay constant k, we can substitute it into the original equation:
M = M₀ * e^(-kt)
M = 50 * e^(-[(ln(2) / 18) * t])
Therefore, the equation that models how much of the sample remains after t years is:
M = 50 * e^(-[(ln(2) / 18) * t])
To find an equation that models how much of the radioactive strontium sample remains after a certain number of years, we can use the concept of exponential decay.
The general formula for exponential decay is given by:
M(t) = M₀ * e^(-kt)
Where:
- M(t) represents the amount of the sample remaining after t years
- M₀ represents the initial amount of the sample
- k is the decay constant
- e is the base of the natural logarithm (approximately 2.71828)
In this case, we are given that the initial amount M₀ is 50 grams and the amount after 18 years (t) is 32 grams. We can use this information to find the value of the decay constant k.
Using the given information, we can plug in the values into the equation:
32 = 50 * e^(-k * 18)
Now, we need to solve for k. You can start by dividing both sides of the equation by 50 to isolate the exponential term:
32/50 = e^(-k * 18)
Now, take the natural logarithm (ln) of both sides to isolate the exponent:
ln(32/50) = -k * 18
Finally, divide both sides by -18 to solve for k:
k = -ln(32/50) / 18
Now that we have the value of k, we can rewrite the equation as:
M(t) = 50 * e^(-[-ln(32/50) / 18] * t)
Simplifying further:
M(t) = 50 * e^(ln(32/50) * t / 18)
M(t) = 50 * e^(ln(32/50) * t / 18)
Now you have the equation that models how much of the sample remains after t years.