The half-life of a certain radioactive material is 38 days. An initial amount of the material has a mass of 497 kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 4 days. Round your answer to the nearest thousandth.
Y=497(1/2)^1/38x;462.029 kg
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I’m taking the test right now and this is my answer
Why did the radioactive material go to therapy? Because it had commitment issues!
Now, let's get to the math. The exponential decay formula is given by:
N(t) = N0 * (1/2)^(t/h).
In this equation:
- N(t) represents the amount of radioactive material remaining at time t.
- N0 is the initial amount of radioactive material.
- t is the time that has elapsed.
- h is the half-life of the material.
Using the information provided, we can plug in the values:
N(t) = 497 * (1/2)^(t/38).
To find out how much radioactive material remains after 4 days, we substitute t = 4 into the equation:
N(4) = 497 * (1/2)^(4/38).
Calculating this expression will give us the value.
To write an exponential function that models the decay of the radioactive material, we can use the formula:
A(t) = A₀ * e^(-kt),
Where:
A(t) is the amount of material remaining after time t,
A₀ is the initial amount of material,
e is the base of the natural logarithm,
k is the decay constant.
Given that the half-life of the material is 38 days, we can use this information to find the decay constant, k:
t₁/2 = ln(2)/k,
38 = ln(2)/k,
k = ln(2)/38.
Now we can substitute the values into the exponential function:
A(t) = 497 * e^((-ln(2)/38)*t).
To find the amount of material remaining after 4 days, we substitute t = 4 into the function:
A(4) = 497 * e^((-ln(2)/38)*4).
Calculating this using a calculator, we find:
A(4) ≈ 497 * e^(-0.018193*4) ≈ 497 * 0.983600 ≈ 488.591.
Therefore, after 4 days, approximately 488.591 kg of radioactive material remains.
To model the decay of the radioactive material, we can use the exponential decay formula:
A(t) = A₀(1/2)^(t/h),
where A(t) represents the amount of material remaining at time t, A₀ is the initial amount of material, t is the time that has passed, and h is the half-life of the material.
In this case, the half-life of the material is 38 days, and the initial amount is 497 kg. So, the exponential decay function that models the decay of this material is:
A(t) = 497(1/2)^(t/38).
To find how much radioactive material remains after 4 days, we substitute t = 4 into the equation:
A(4) = 497(1/2)^(4/38).
Now we can evaluate this expression to find the answer.
A(4) = 497(0.985894),
A(4) ≈ 491.46 kg.
Therefore, after 4 days, approximately 491.46 kg of the radioactive material remains.
A(t) = 497*(1/2)^(t/38)
You can see that when t grows by 38, you multiply again by 1/2
now plug in t=4