A half life of a certain radioactive is 36 days. An initial amount of the material has a mass of 487 kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 5 days. Round your answer to the nearest thousandth.

amount = 487 (1/2)^(t/36) where t is the number of days.

if t = 5
amount = 487(.5)^(5/36)
= 442.302 kg

Why did the radioactive material go on a diet? Because it wanted to have a smaller mass!

Now, let's get serious and calculate how much radioactive material remains after 5 days using an exponential function.

The general formula for exponential decay is A(t) = A₀ * e^(-kt), where A(t) is the amount of material remaining at time t, A₀ is the initial amount of material, e is Euler's number (approximately 2.71828), and k is the decay constant.

To find the decay constant, we can use the formula k = ln(2) / T½, where ln is the natural logarithm and T½ is the half-life of the material.

In this case, the half-life is 36 days, so the decay constant is k = ln(2) / 36.

Now, substituting these values into our decay formula, we have A(t) = 487 * e^(-t * ln(2) / 36).

To find how much material remains after 5 days, we can plug in t = 5 into the formula:

A(5) = 487 * e^(-5 * ln(2) / 36).

Using a calculator, we find that A(5) is approximately 454.026 kg.

So, after 5 days, approximately 454.026 kg of radioactive material remains.

To write an exponential function that models the decay of the radioactive material, we can use the formula:

\[N(t) = N_0 \times e^{-kt}\]

Where:

- \(N(t)\) is the remaining amount of radioactive material at time \(t\).
- \(N_0\) is the initial amount of the material.
- \(k\) is the decay constant.

The decay constant, \(k\), can be calculated using the formula:

\[k = \frac{ln(2)}{t_{\frac{1}{2}}}\]

Where:

- \(ln\) is the natural logarithm.
- \(t_{\frac{1}{2}}\) is the half-life of the material.

In this case, the half-life (\(t_{\frac{1}{2}}\)) is 36 days.

Plugging these values into the formula, we get:

\[k = \frac{ln(2)}{36}\]

Now we can write the exponential function:

\[N(t) = 487 \times e^{-\frac{ln(2)}{36} \times t}\]

To find how much radioactive material remains after 5 days, we substitute \(t = 5\) into the function:

\[N(5) = 487 \times e^{-\frac{ln(2)}{36} \times 5}\]

Calculating this expression will give us the answer rounded to the nearest thousandth.

To write an exponential function that models the decay of the radioactive material, we can use the general form of an exponential decay function:

N(t) = N(0) * (1/2)^(t/h)

Where:
- N(t) is the amount of material remaining after time t
- N(0) is the initial amount of material
- t is time
- h is the half-life of the material

Given that the half-life of the material is 36 days and the initial amount of material is 487 kg, the exponential decay function for this situation is:

N(t) = 487 * (1/2)^(t/36)

To find out how much radioactive material remains after 5 days, we substitute t = 5 into the equation:

N(5) = 487 * (1/2)^(5/36)

Calculating this expression, we get N(5) = 487 * (0.942) ≈ 458.994

Therefore, after 5 days, approximately 459 kg of radioactive material remains.