Following surgery, a patient has been receiving a pain relief medication intravenously. The concentration $C$ (in milligrams per liter) of the medication in the patient's bloodstream $t$ hours after this process started is given by
\[
C(t)=\frac{40 t+8}{t+4}, t \geq 0
\]
a) The patient will not receive pain relief unless the concentration of the medication is 20 or more milligrams per liter. Use this function to determine the time interval for which the concentration of the medication, $C$, will be greater than or equal to 20 milligrams per liter.
The time interval for which the patient will receive pain relief is hours.
Answer
\(\boxed{t \geq 3.6}\)
Step 1 ：\(\frac{40 t+8}{t+4} \geq 20\)
Step 2 ：Multiply both sides by \((t+4)\) to get \(40 t+8 \geq 20(t+4)\)
Step 3 ：Simplify to get \(40 t+8 \geq 20t+80\)
Step 4 ：Subtract \(20t\) from both sides to get \(20t+8 \geq 80\)
Step 5 ：Subtract 8 from both sides to get \(20t \geq 72\)
Step 6 ：Divide both sides by 20 to get \(t \geq 3.6\)
Step 7 ：\(\boxed{t \geq 3.6}\)
