an element with a mass of 520 grams decays by 25.1% per minute. to the nearest tenth of a minute, how long…

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, $A_0=520$, $r = 0.251$, and $A = 60$. So the equation becomes $60=520(1 - 0.251)^t$, which simplifies to $60 = 520\times0.749^t$.

Step2: Isolate the exponential term

Divide both sides of the equation by 520: $\frac{60}{520}=0.749^t$, so $0.1154 = 0.749^t$.

Step3: Take the natural - logarithm of both sides

$\ln(0.1154)=\ln(0.749^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(0.1154)=t\ln(0.749)$.

Step4: Solve for $t$

$t=\frac{\ln(0.1154)}{\ln(0.749)}$. We know that $\ln(0.1154)\approx - 2.16$ and $\ln(0.749)\approx-0.29$. Then $t=\frac{- 2.16}{-0.29}\approx7.4$.

Answer:

$7.4$