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 resect 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.

please could someone answer this question!

To determine the average rate of change of the amount of medicine ingested during each interval, we need to find the change in the amount of medicine (A) over the given time interval.

a) For the interval from 4 to 6 hours:
Average rate of change = (Change in A) / (Change in t)

To find the change in A, we substitute t = 6 and t = 4 into the function A(t):
A(6) = (5/8)(12(6)-(6)^2) = 45 mg
A(4) = (5/8)(12(4)-(4)^2) = 64 mg

Change in A = A(6) - A(4) = 45 - 64 = -19 mg

Change in t = 6 - 4 = 2 hours

Average rate of change = (-19 mg) / (2 hours) = -9.5 mg/hour

For the interval from 6 to 8 hours:
Average rate of change = (Change in A) / (Change in t)

To find the change in A, we substitute t = 8 and t = 6 into the function A(t):
A(8) = (5/8)(12(8)-(8)^2) = 32 mg
A(6) = (5/8)(12(6)-(6)^2) = 45 mg

Change in A = A(8) - A(6) = 32 - 45 = -13 mg

Change in t = 8 - 6 = 2 hours

Average rate of change = (-13 mg) / (2 hours) = -6.5 mg/hour

Comparing the two rates:
The average rate of change from 4 to 6 hours is -9.5 mg/hour, while the average rate of change from 6 to 8 hours is -6.5 mg/hour. Therefore, during the first interval, the amount of medicine ingested is decreasing at a faster rate compared to the second interval.

b) To estimate the slope of the tangent at the point P(4, 80/3), we can find the derivative of the function A(t) and evaluate it at t = 4:

A'(t) = (5/8)(12 - 2t)

Evaluating at t = 4:
A'(4) = (5/8)(12 - 2(4)) = (5/8)(12 - 8) = (5/8)(4) = 5/2 = 2.5

Therefore, the slope of the tangent at the point P(4, 80/3) is approximately 2.5.

For this situation, the value represents the instantaneous rate of change of the amount of medicine ingested at 4 hours. It indicates the speed at which the amount of medicine is being ingested at that specific time.