Question 538837
"A patient weighing 150 pounds requires a dose of 850 milligrams, while a patient weighing 300 pounds requires a dose of 1700 milligrams."
They seem to be hinting that the dose needed varies directly with the weight of the patient, and that we can calculate the dose by using the factor
{{{1700mg/300lb=850mg/150lb=(17/3)(mg/lb)}}} about 5.67 mg/lb or 12.5 mg/kg.
The linear function they want in 1) is the line we would get from the two points given (the 150 pound patient and the 300 pound patient):
{{{dose=(17/3)(mg/lb)}}}
NOTE
Math nerds like me do not know that the relationship would be lineal, but I trained/worked in pharmacology and know that doses are often expressed per kilogram body weight (even when working with 20-gram mice), so direct relations are assumed.
2) The slope is the dose per pound ratio that you would use. You would multiply patient weight times that slope.
EXTRA INSIGHTS
Nowadays scales in hospitals give weight in kilograms and the nurse translates the reading into pounds for the patient's benefit (not that it will necessarily make the patient happy). Since the problem refers to pounds, this is an old problem or not a realistic one.
A 240 pound patient would need 1360 mg chloramphenicol {{{(17/3)*240=1360}}}, but you did not ask that question.
The equation {{{C (t)=83.3t^2 + 583.1t-170.4}}} does not make sense. It would mean that the patient's blood level would start negative (OK, maybe it's only a good model after half an hour or so), and would continue to increase forevermore.
If the blood levels would vary as {{{C (t)=-83.3t^2 + 583.1t-170.4}}} they would be 125 mg/mL when {{{-83.3t^2 + 583.1t-295.4=0}}}, and
{{{t = (-583.1 +- sqrt( 583.1^2-4*(-83.3)*(-295.4) ))/(2*(-83.3))=(583.1 +- sqrt( 340006-98427))/166.6=(583.1 +- sqrt(241578))/166.6=(583.1 +- 491.5)/166.6}}}
That would give you t=0.55 hours and t=6.45 hours.
I would give a dose every 6 hours, but you don't ask about that either.
3)"The doctor wishes to use a 10% solution of Chloramphenicol, administered through an intravenous injection. The hospital only carries 5% and 20% solutions of the required medication."
1000mL of a 10% solution would contain {{{0.10*1000=100}}} grams.
The amount in x mL of 5% solution would be {{{0.05x}}} grams.
The amount in (1000-x) mL of 20% solution would be {{{0.20(1000-x)}}} grams.
So {{{0.05x+0.20(1000-x)=100}}} ---> {{{0.05x+200-0.20x=100}}} ---> {{{200-0.15x=100}}} ---> {{{100=0.15x}}} ---> {{{x=100/0.15}}} ---> {{{x=667}}}
You would use 667 mL of 5% solution and 333 mL of 20% solution.