The amount of active ingredient of medicine ingested by the body, A in milligrams, is a function of time, t, in hours, given by A(t)=5/8(12t-t^2).
a)Determine the average rate of change of the amount of medicine ingested during each of the intervals below and compare the two rates with respect to the given situation.
4 to 6 hours and 6 to 8 hours.
b) Estimate the slope of the tangent at the point P(4,80/3). Use interval of 0.01. Round final answer to the nearest hundredth. Explain what the value represents for this situation.
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To answer these questions, we need to understand the concepts of average rate of change and tangent slope.
a) Average rate of change:
The average rate of change of a function over an interval is calculated by finding the difference in the function values at the endpoints of the interval and dividing it by the difference in the independent variable (in this case, time) at the endpoints.
For the interval from 4 to 6 hours:
A(4) = (5/8)(12(4)-(4^2)) = (5/8)(48-16) = (5/8)(32) = 20 units
A(6) = (5/8)(12(6)-(6^2)) = (5/8)(72-36) = (5/8)(36) = 22.5 units
The average rate of change over this interval is ((22.5-20)/(6-4)) = (2.5/2) = 1.25 units/hour.
For the interval from 6 to 8 hours:
A(6) = 22.5 units
A(8) = (5/8)(12(8)-(8^2)) = (5/8)(96-64) = (5/8)(32) = 20 units
The average rate of change over this interval is ((20-22.5)/(8-6)) = (-2.5/2) = -1.25 units/hour.
Comparing the two rates, we can see that the rate of change of the amount of medicine ingested decreases from 4 to 6 hours to 6 to 8 hours.
b) Estimating the slope of the tangent at the point P(4, 80/3):
To estimate the slope of the tangent at a specific point, we use an infinitesimally small interval around that point and calculate the average rate of change over that interval.
Let's estimate the slope using an interval of 0.01 around point P(4, 80/3):
A(4) = 80/3 units
A(4.01) = (5/8)(12(4.01)-(4.01^2)) ≈ 27.074 units
The average rate of change over this interval is approximately ((27.074 - (80/3))/(4.01 - 4)) ≈ -20.51 units/hour.
Hence, the estimated slope of the tangent at point P(4, 80/3) is approximately -20.51 units/hour.
The value represents the rate at which the amount of medicine ingested is changing at precisely that moment in time. In this case, it indicates the rate at which the amount of medicine ingested is decreasing at 4 hours.