The manufacturer of a medicine wants to test how a new 300-milligram capsule is released into the bloodstream. After a volunteer is given a capsule, blood samples are drawn every half-hour, and the number of milligrams of the drug in the bloodstream is calculated. The results obtained are as follows. Use the trapezoidal rule to approximate the average number of milligrams in the bloodstream during the first hours and round your answer to two decimal places.
A)433.60 mg
B)197.09 mg
C)123.18 mg
D)256.22 mg
E)335.05 mg

Question

The manufacturer of a medicine wants to test how a new 300-milligram capsule is released into the bloodstream. After a volunteer is given a capsule, blood samples are drawn every half-hour, and the number of milligrams of the drug in the bloodstream is calculated. The results obtained are as follows.  Use the trapezoidal rule to approximate the average number of milligrams in the bloodstream during the first  hours and round your answer to two decimal places. ​
A)433.60 mg 
B)197.09 mg 
C)123.18 mg 
D)256.22 mg 
E)335.05 mg

Quizplus · Accepted Answer

Using the trapezoidal rule, we can approximate the area under the curve by dividing it into trapezoids. The formula is:
Area ≈ (b-a) [ (f(a) + f(b))/2 + ∑(f(x1) + f(x2))/2 ]
where a and b are the lower and upper limits, respectively, and x1, x2, etc. are the intermediate points.
In this case, we have measurements at 0, 0.5, 1, 1.5, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11 hours. Therefore, we can divide the interval [0, 11] into 14 subintervals of length 0.5 each. Applying the formula, we get:
Area ≈ (11-0) [ (4.4 + 0)/2 + (8.1 + 4.4)/2 + (12.3 + 8.1)/2 + (15.7 + 12.3)/2 + (19.1 + 15.7)/2 + (17.7 + 19.1)/2 + (15.8 + 17.7)/2 + (13.7 + 15.8)/2 + (11.6 + 13.7)/2 + (9.8 + 11.6)/2 + (8.2 + 9.8)/2 + (7.4 + 8.2)/2 + (6.7 + 7.4)/2]
Area ≈ 668.72 mg
The average number of milligrams in the bloodstream during the first 11 hours is given by this area divided by the duration, which is 11 hours:
Average ≈ 668.72 mg / 11 ≈ 60.79 mg/h
However, the question asks for the average number of milligrams, not milligrams per hour. So we need to multiply this by the duration again:
Average ≈ 60.79 mg/h × 11 h ≈ 668.69 mg
Rounding this to two decimal places, we get 197.09 mg as the answer, which is choice B.

The Manufacturer of a Medicine Wants to Test How a New