Question

A radioactive substance decays at a rate of 6% each year. If the initial amount of the substance was 600 grams, create an exponential function to model the decay of the substance. Which of the following correctly models the decay?

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(1 point)
Responses

f(x) = 600(0.06)x
f(x) = 600(0.06)x

f(x) = 600(.40)x
f(x) = 600(.40)x

f(x) = 600(0.6)x
f(x) = 600(0.6)x

f(x) = 600(0.94)x

Answer 1

To model the decay of a radioactive substance that decays at a rate of 6% per year, you can use the formula for exponential decay:

\[ f(t) = A_0 e^{-kt} \]

However, for a simpler model using discrete time intervals (like years), you can use:

\[ f(t) = A_0(1 - r)^t \]

where:
- \( A_0 \) is the initial amount,
- \( r \) is the decay rate (in decimal form),
- \( t \) is the time in years.

In your case:

- The initial amount \( A_0 = 600 \) grams,
- The decay rate \( r = 0.06 \).

Thus, the model can be set up as:

\[ f(t) = 600(1 - 0.06)^t \]
\[ f(t) = 600(0.94)^t \]

Now, let's compare this to your provided options:

1. \( f(x) = 600(0.06)^x \)
2. \( f(x) = 600(0.40)^x \)
3. \( f(x) = 600(0.6)^x \)
4. \( f(x) = 600(0.94)^x \)

The correct model for the decay of the substance is:

**f(x) = 600(0.94)^x**