Question

Please answer these two questions:

1. The amount A, in milligrams, of a 10-milligram dose of a drug remaining in the body
reduces at a rate of 20%. Find, to the nearest tenth of an hour, how long it takes for
half of the drug dose to be left in the body.

2. After t years, the rate of depreciation of a car that costs $20,000 is 25%. What is the value of the car 2 years after it was purchased?

show work please!!

Answer 1

so if the inital amount in the body is 10mg, so half that will just be 5mg, so how long will that be?

`\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\dotfill & 5~mg\\ P=\textit{initial amount}\dotfill &10~mg\\ r=rate\to 20\%\to (20)/(100)\dotfill &0.2\\ t=hours\dotfill &t\\ \end{cases}`

`5 = 10(1 - 0.2)^(t) \implies \cfrac{5}{10}=0.8^t\implies \cfrac{1}{2}=0.8^t\implies \log\left( \cfrac{1}{2} \right)=\log(0.8^t) \\\\\\ \log\left( \cfrac{1}{2} \right)=t\log(0.8)\implies \cfrac{\log\left( (1)/(2) \right)}{\log(0.8)}=t\implies \stackrel{ \textit{about 3 hrs and 6 mins} }{3.1\approx t}`