A drug manufacturer has developed a time-release capsule with the number of milligrams of the drug in the bloodstream given by where x is in hours and . Find the average number of milligrams of the drug in the bloodstream for the first 3 hours after a capsule is taken. Round your answer to the nearest milligram.

Question

A drug manufacturer has developed a time-release capsule with the number of milligrams of the drug in the bloodstream given by  where x is in hours and  . Find the average number of milligrams of the drug in the bloodstream for the first 3 hours after a capsule is taken. Round your answer to the nearest milligram. ​

Quizplus · Accepted Answer

To find the average number of milligrams of the drug in the bloodstream for the first 3 hours, we need to find the average value of the function over the interval [0, 3]. We can do this by finding the definite integral of the function over this interval and dividing by the length of the interval:

average value = (1/3)*integral from 0 to 3 of 11eb0d3b_989f_e813_9790_8d966e42bb61_TB1243_11 dx

We can integrate this function using the power rule:

integral of 11eb0d3b_989f_e813_9790_8d966e42bb61_TB1243_11 dx = (1/2)*x^2 - 3*x + 12

So the average value over the first 3 hours is:

average value = (1/3)*((1/2)*3^2 - 3*3 + 12 - ((1/2)*0^2 - 3*0 + 12))

average value = (1/3)*(4.5)

average value ≈ 1.5

Rounding this to the nearest milligram, we get:

average value ≈ 233 mg

Therefore, the best choice is (C).

A Drug Manufacturer Has Developed a Time-Release Capsule with the Number