question\nan element with a mass of 310 grams...

# Explanation: ## Step1: Set up the decay formula The formula for exponential decay is $A = A_0(1 - r)^t$, where $A$ is the final amount, $A_0$ is the initial amount, $r$ is the rate of decay, and $t$ is the time. Here, $A_0=310$, $r = 0.064$, and $A = 100$. So the equation becomes $100=310(1 - 0.064)^t$, which simplifies to $100 = 310\times0.936^t$. ## Step2: Rearrange the equation Divide both sides of the equation by 310: $\frac{100}{310}=0.936^t$, or $0.3226 = 0.936^t$. ## Step3: Take the natural - logarithm of both sides $\ln(0.3226)=\ln(0.936^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(0.3226)=t\ln(0.936)$. ## Step4: Solve for $t$ $t=\frac{\ln(0.3226)}{\ln(0.936)}$. We know that $\ln(0.3226)\approx - 1.133$ and $\ln(0.936)\approx-0.066$. Then $t=\frac{- 1.133}{-0.066}\approx17.2$. # Answer: $17.2$