an element with a mass of 200 grams decays by 15.7% per minute. to the nearest minute, how long will it be…

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, $A_0=200$, $r = 0.157$, and $A = 60$. So the equation becomes $60=200(1 - 0.157)^t$, which simplifies to $60 = 200\times0.843^t$.

Step2: Rearrange the equation

Divide both sides of the equation $60 = 200\times0.843^t$ by 200: $\frac{60}{200}=0.843^t$, so $0.3 = 0.843^t$.

Step3: Take the natural - logarithm of both sides

$\ln(0.3)=\ln(0.843^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(0.3)=t\ln(0.843)$.

Step4: Solve for $t$

$t=\frac{\ln(0.3)}{\ln(0.843)}$. Calculate $\ln(0.3)\approx - 1.204$ and $\ln(0.843)\approx- 0.169$. Then $t=\frac{-1.204}{-0.169}\approx7.124$.

Answer:

7