Question

1. krypton - 91 is a radioactive substance that decays very quickly. the function q(t)=q₀e⁻ᵏᵗ models the radioactive decay of krypton - 91. q represents the quantity remaining after t seconds, and the decay constant, k, is approximately 0.07. how long will it take a quantity of krypton - 91 to decay to 10% of its original amount? round your answer to the nearest second.(3 points)

Explanation:

Step1: Set up the equation

Let the initial amount be $Q_0$. We want $Q(t)=0.1Q_0$. Substitute into the decay - formula $Q(t)=Q_0e^{-kt}$. So, $0.1Q_0 = Q_0e^{-0.07t}$.

Step2: Simplify the equation

Divide both sides of the equation $0.1Q_0 = Q_0e^{-0.07t}$ by $Q_0$ (since $Q_0
eq0$), we get $0.1 = e^{-0.07t}$.

Step3: Take the natural - logarithm of both sides

$\ln(0.1)=\ln(e^{-0.07t})$. Using the property $\ln(e^x)=x$, the right - hand side simplifies to $- 0.07t$. So, $\ln(0.1)=-0.07t$.

Step4: Solve for $t$

We know that $\ln(0.1)\approx - 2.3026$. Then, $t=\frac{\ln(0.1)}{-0.07}=\frac{-2.3026}{-0.07}\approx33$.

Answer:

33