Question

question 12 of 24 step 3 of 3 the half - life of gold - 194 is approximately 1.6 days. step 3 of 3: how much of a 9 gram sample of gold - 194 would remain after 4 days? round to three decimal places. answer how to enter your answer (opens in new window) 2 points

Explanation:

Step 1: Recall the half - life formula

The formula for radioactive decay is $N(t)=N_0\times(\frac{1}{2})^{\frac{t}{T_{1/2}}}$, where $N(t)$ is the amount of the substance remaining after time $t$, $N_0$ is the initial amount of the substance, and $T_{1/2}$ is the half - life of the substance.
Here, $N_0 = 9$ grams, $t = 4$ days, and $T_{1/2}=1.6$ days.

Step 2: Substitute the values into the formula

First, calculate the exponent $\frac{t}{T_{1/2}}=\frac{4}{1.6}=\frac{40}{16}=\frac{5}{2} = 2.5$.
Then, $N(4)=9\times(\frac{1}{2})^{2.5}$.
We know that $(\frac{1}{2})^{2.5}=2^{- 2.5}=\frac{1}{2^{2.5}}$. Since $2^{2.5}=2^{2 + 0.5}=2^{2}\times2^{0.5}=4\sqrt{2}\approx4\times1.4142 = 5.6568$. So, $(\frac{1}{2})^{2.5}=\frac{1}{5.6568}\approx0.1768$.
Then $N(4)=9\times0.1768 = 1.5912$. Rounding to three decimal places, we get $N(4)\approx1.591$.

Answer:

1.591