Question

C. How long until there are only 20 mg remaining.

Answer 1

`\begin{gathered} \text{Given} \\ C(t)=85\mleft((1)/(2)\mright)^{(t)/(6)} \end{gathered}`

`\begin{gathered} \text{C. How long until there are only 20 mg remaining.} \\ \\ \text{Let }C(t)=20,\text{ then solve for }t \\ \\ C(t)=85\mleft((1)/(2)\mright)^{(t)/(6)} \\ 20=85\mleft((1)/(2)\mright)^{(t)/(6)} \\ \\ \text{Divide both sides by }85 \\ (20)/(85)=\frac{85\mleft((1)/(2)\mright)^{(t)/(6)}}{85} \\ (20)/(85)=\mleft((1)/(2)\mright)^{(t)/(6)} \\ \\ \text{Get the natural logarithm of both sides} \\ \mleft((1)/(2)\mright)^{(t)/(6)}=(20)/(85) \\ \ln \mleft((1)/(2)\mright)^{(t)/(6)}=\ln (20)/(85) \\ (t)/(6)\ln \mleft((1)/(2)\mright)^{}=\ln (20)/(85) \\ \\ \text{Multiply both sides by }(6)/(\ln (1)/(2)) \\ (6)/(\ln(1)/(2))\Big[(t)/(6)\ln ((1)/(2))^{}=\ln (20)/(85)\Big](6)/(\ln(1)/(2)) \\ \frac{\cancel{6}}{\cancel{\ln (1)/(2)}}\Big{[}\frac{t}{\cancel{6}}\cancel{\ln ((1)/(2))}^{}=\ln (20)/(85)\Big{]}(6)/(\ln(1)/(2)) \\ \\ t=(6\ln (20)/(85))/(\ln (1)/(2)) \\ t=12.52477705 \end{gathered}`

Therefore, it will take 12.52 hours