Solve the problem.
-The function $\mathrm { A } = \mathrm { A } _ { 0 } \mathrm { e } ^ { - 0.01155 \mathrm { x } }$ models the amount in pounds of a particular radioactive material stored in a concrete vault, where $x$ is the number of years since the material was put into the vault. If 500 pounds of the material are placed in the vault, how much time will need to pass for only 140 pounds to remain?
A) 110 years
B) 115 years
C) 120 years
D) 220 years

Question

Quizplus · Accepted Answer

The amount of radioactive material remaining after $x$ years is given by the function $\mathrm { A } = \mathrm { A } _ { 0 } \mathrm { e } ^ { - 0.01155 \mathrm { x } }$. If 500 pounds of the material are placed in the vault, then $\mathrm { A } _ { 0 } = 500$. If only 140 pounds remain, then $\mathrm { A } = 140$. Substituting these values into the function, we get:$$140 = 500\mathrm { e } ^ { - 0.01155 \mathrm { x } }$$Dividing both sides by 500, we get:$$0.28 = \mathrm { e } ^ { - 0.01155 \mathrm { x } }$$Taking the natural logarithm of both sides, we get:$$\ln(0.28) = - 0.01155 \mathrm { x }$$Solving for $x$, we get:$$x = \frac{\ln(0.28)}{-0.01155} \approx 110$$Therefore, it will take approximately 110 years for only 140 pounds of the radioactive material to remain.

Solve the Problem A=A0e−0.01155x\mathrm { A } = \mathrm { A } _ { 0 } \mathrm { e } ^ { - 0.01155 \mathrm { x } }A=A0​e−0.01155x

Solve the Problem $\mathrm { A } = \mathrm { A } _ { 0 } \mathrm { e } ^ { - 0.01155 \mathrm { x } }$