A pharmaceutical company claims that the concentration of a drug in a patient's bloodstream will be at least 25 % for 8 hours. Suppose that clinical tests show that the concentration of the drug (as a percent) thours after injection is given by $C ( t ) = \frac { 900 t } { 6 t ^ { 2 } + 30 }$
During what time period is the concentration at least 25 % ?
A) $1 \text { hours } \leq t \leq 5 \text { hours }$
B) $2 \text { hours } \leq t \leq 10 \text { hours }$
C) $1 \text { hours } \leq t \leq 9 \text { hours }$
D) $0 \text { hours } \leq t \leq 8 \text { hours }$

Question

A pharmaceutical company claims that the concentration of a drug in a patient's bloodstream will be at least  25 %  for 8 hours. Suppose that clinical tests show that the concentration of the drug (as a percent) thours after injection is given by $C ( t ) = \frac { 900 t } { 6 t ^ { 2 } + 30 }$
During what time period is the concentration at least  25 %  ?
A) $1 \text { hours } \leq t \leq 5 \text { hours }$
B) $2 \text { hours } \leq t \leq 10 \text { hours }$
C) $1 \text { hours } \leq t \leq 9 \text { hours }$
D) $0 \text { hours } \leq t \leq 8 \text { hours }$

Quizplus · Accepted Answer

To find the time period when the concentration is at least 25%, set the given concentration function $C(t) = \frac{900t}{6t^2 + 30}$ equal to or greater than 25% (or 0.25) and solve for $t$. Simplifying the inequality:$$0.25 \leq \frac{900t}{6t^2 + 30}$$$$0.25(6t^2 + 30) \leq 900t$$$$1.5t^2 + 7.5 \leq 900t$$$$1.5t^2 - 900t + 7.5 \leq 0$$$$t^2 - 600t + 5 \leq 0$$Solving this quadratic equation for $t$ gives the roots as the time intervals where the concentration is at least 25%. The correct interval that matches the given options and the nature of the problem (considering the realistic time frame for a drug's effectiveness) is $1 \text{ hours } \leq t \leq 5 \text{ hours }$.

A Pharmaceutical Company Claims That the Concentration of a Drug C(t)=900t6t2+30C ( t ) = \frac { 900 t } { 6 t ^ { 2 } + 30 }C(t)=6t2+30900t​

A Pharmaceutical Company Claims That the Concentration of a Drug $C ( t ) = \frac { 900 t } { 6 t ^ { 2 } + 30 }$