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 options correctly models the decay?

Option #1: f(x) = 600 (0.06)x

Option #2: f(x) = 600(1.06)x


Option #3: f(x) = 600(0.6)x


Option #4: f(x) = 600(0.94)x

The option that correctly models the problem is Option #

Answer 1

To model the decay of a radioactive substance, we typically use the formula:

\[
f(t) = f_0 \cdot (1 - r)^t
\]

where:
- \( f(t) \) is the amount of substance at time \( t \),
- \( f_0 \) is the initial amount,
- \( r \) is the decay rate (as a decimal),
- \( t \) is the time in years.

Given that the decay rate is 6%, we can express this as \( r = 0.06 \). The initial amount is 600 grams.

Therefore, the function for this decay would be:

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

Now, let's look at the options provided:

- Option #1: \( f(x) = 600 (0.06)^x \) (This is incorrect as this represents a growth rather than decay.)
- Option #2: \( f(x) = 600(1.06)^x \) (This is incorrect as this also represents growth rather than decay.)
- Option #3: \( f(x) = 600(0.6)^x \) (This is incorrect as this does not represent the given decay rate.)
- Option #4: \( f(x) = 600(0.94)^x \) (This is correct as it represents the 6% decay.)

Thus, the option that correctly models the problem is **Option #4**: \( f(x) = 600(0.94)^x \).