Question

The mass of a radioactive chemical element in a sample decreases over time. The relationship between the elapsed time t, in weeks, since a sample of the element was measured, and its mass, M(t), in grams is modeled by the following function:

M t = (0.89)^5t+3

What factor is the mass of the sample multiplied by every week

Answer 1

To determine the factor by which the mass of the sample is multiplied every week, we can analyze the given function:

\[ M(t) = (0.89)^{5t + 3} \]

Let's express \( M(t) \) in the context of weekly decay. First, we can rewrite the function in terms of \( t \):

\[
M(t) = (0.89)^{5t + 3} = (0.89)^3 \cdot (0.89)^{5t}
\]

Now, we can see that \( (0.89)^{3} \) is a constant factor that does not change as \( t \) increases. The term \((0.89)^{5t}\) represents the decay over time.

To find the decay factor per week, we need to isolate the factor for one week. Setting \( t \) to \( 1 \) gives us:

\[
M(1) = (0.89)^{5(1) + 3} = (0.89)^{5 + 3} = (0.89)^{8}
\]

So, the mass after one week becomes:

\[
M(1) = M(0) \cdot (0.89)^{5}
\]

If we consider the general form of decay, every week would see the mass multiplied by a constant factor determined by the exponent. The given function \( (0.89)^{5t} \) indicates that for each increase by \( t \) (in units of time that were scaled), the mass changes by a factor of

\[
(0.89)^{5}
\]

over the period corresponding to \( t \).

To find the weekly factor \( R \), calculate:

\[
R = (0.89)^{5}
\]

Calculating this gives:

\[
R \approx 0.5277
\]

Thus, the mass of the sample is multiplied by approximately **0.5277** every week, indicating that the mass decreases to about **52.77%** of the previous week's mass.

In summary, every week, the mass is multiplied by approximately **0.5277**.