uranium - 238 (u - 238) has a half - life of 4.5 billion years. geologists find a rock containing a mixture…

Explanation:

Step1: Recall the radioactive - decay formula

The formula for radioactive decay is $N = N_0\left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$, where $N$ is the amount of the radioactive substance at time $t$, $N_0$ is the initial amount of the radioactive substance, $T_{1/2}$ is the half - life of the radioactive substance. We know that $\frac{N}{N_0}=0.65$ (since 65% of the original U - 238 remains) and $T_{1/2}=4.5$ billion years.

Step2: Substitute values into the formula

Substitute $\frac{N}{N_0}=0.65$ and $T_{1/2}=4.5$ into the formula $N = N_0\left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$, we get $0.65=\left(\frac{1}{2}\right)^{\frac{t}{4.5}}$.

Step3: Take the natural logarithm of both sides

$\ln(0.65)=\ln\left(\left(\frac{1}{2}\right)^{\frac{t}{4.5}}\right)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we have $\ln(0.65)=\frac{t}{4.5}\ln\left(\frac{1}{2}\right)$.

Step4: Solve for $t$

We know that $\ln(0.65)\approx - 0.4308$ and $\ln\left(\frac{1}{2}\right)=-\ln(2)\approx - 0.6931$. Then $t = 4.5\times\frac{\ln(0.65)}{\ln\left(\frac{1}{2}\right)}$. $t = 4.5\times\frac{- 0.4308}{- 0.6931}\approx2.797$

Answer:

$2.797$