h) Exponential functions can be used to model this situation, as the concentration of penicillin in the body decreases exponentially over time.
d) Let C(t) be the concentration of penicillin in the body at time t in hours after the initial dose. Then, the equation to represent this situation is:
C(t) = 500(0.8)^t
This equation represents an exponential decay function, where the initial concentration of 500 milligrams decreases by a factor of 0.8 for each hour that passes.
j) In order for the concentration of penicillin in the body to remain greater than 100 milligrams, we can set up the following inequality:
C(t) > 100
Substituting the equation from part d), we get:
500(0.8)^t > 100
Dividing both sides by 500, we get:
(0.8)^t > 0.2
Taking the logarithm base 0.8 of both sides, we get:
t > log0.8(0.2)
Solving for t, we get:
t > 6.9078
Therefore, the medicine needs to be taken at least every 7 hours in order to maintain a concentration greater than 100 milligrams.
Connections: Both situations can be modeled using exponential functions, with the first situation representing exponential growth and the second situation representing exponential decay. In both cases, the initial value decreases or increases by a certain factor over time. The equations, tables, and graphs for both situations will have similar shapes, with the main difference being the direction of the curve (upward or downward). Additionally, both situations involve maintaining a certain threshold value, which can be represented by an inequality.