Both antibiotics were prescribed in high dosage slow release capsules. The function C(x)=5log(x+1)+10 models the concentration of levofloxacin in mol/L over a time x, in hours. The function D (x)=10log(x+1)+5 models the concentration of metronidazole in mol/L over a time x, in hours.
A) Which of the two drugs has a higher initial concentration in the blood
stream? Justify your answer with an explanation.
B)Determine when C(x)=D(x) algebraically and state what this
represents in this situation.
C)If Mathews is instructed to take both antibiotics at once the
concentration levels could be modeled by the function (C+D)(x) . How would the graph of (C+D)(x) differ from the individual graphs of C(x) and D(x)? explain.
(A) what are the values when t=0?
(B) so do the math
5log(x+1)+10 = 10log(x+1)+5
log(x+1) = 1
now finish it off
(C) the graph of (C+D)(x) is just the sum of C(x)+D(x)
there is no value for t
oh, well, duh. then when x=0
A) To determine which drug has a higher initial concentration in the bloodstream, we can compare their respective functions C(x) and D(x) at x = 0 (initial time).
For C(x), substituting x = 0 into the equation:
C(0) = 5log(0+1) + 10
C(0) = 5log(1) + 10
C(0) = 5(0) + 10
C(0) = 10
Similarly, for D(x), substituting x = 0 into the equation:
D(0) = 10log(0+1) + 5
D(0) = 10log(1) + 5
D(0) = 10(0) + 5
D(0) = 5
Comparing the values, we see that C(0) = 10 and D(0) = 5. Therefore, the drug with a higher initial concentration in the bloodstream is levofloxacin (C(x)), which has an initial concentration of 10 mol/L, while metronidazole (D(x)) has an initial concentration of 5 mol/L.
B) To determine when C(x) = D(x), we can set the two equations equal to each other and solve for x algebraically:
5log(x + 1) + 10 = 10log(x + 1) + 5
By subtracting 5log(x + 1) from both sides and rearranging, we get:
5 = 5log(x + 1)
Dividing both sides by 5:
1 = log(x + 1)
Since logarithm functions can be rewritten in exponential form, we can rewrite the equation as:
10^1 = x + 1
Simplifying further:
10 = x + 1
x = 9
Therefore, C(x) = D(x) when x = 9. In this situation, x = 9 represents the number of hours when the concentrations of levofloxacin and metronidazole in the bloodstream are equal.
C) The graph of (C+D)(x) represents the combined concentration of both antibiotics when taken at once. To understand how it differs from the individual graphs of C(x) and D(x), we can consider the properties of logarithmic functions.
The addition of C(x) and D(x) results in the function (C+D)(x), which implies that the concentrations of both drugs are added together at each point in time. This means that (C+D)(x) will have higher concentration values than the individual graphs of C(x) and D(x) at each corresponding time.
Graphically, this means that the graph of (C+D)(x) will be higher than both the graph of C(x) and D(x) at all points on the x-axis. The rates of increase for (C+D)(x) will be faster than the rates of increase for C(x) and D(x) individually.
Essentially, taking both antibiotics at once leads to a higher overall concentration in the bloodstream compared to taking them individually. This is due to the combined effect of the logarithmic functions when added together.