Answer:
Step-by-step explanation:
Functions
The problem describes a function that expresses the concentration of an antibiotic in mg/dl vs time in hours as:
[tex]\displaystyle c(t)=\frac{50\ t}{t^2+25}[/tex]
We need to find the first value of t such that
[tex]\displaystyle c(t)\geq 4[/tex]
It means that
[tex]\displaystyle \frac{50\ t}{t^2+25}\geq 4[/tex]
Operating with the inequality
[tex]\displaystyle 50\ t\geq 4\ t^2+100[/tex]
Rearranging and dividing by 2, we have a polynomial inequality:
[tex]\displaystyle 2t^2-25t+50\leq 0[/tex]
Factoring
[tex]\displaystyle 2(t-10)\left (t-\frac{5}{2}\right )\leq 0[/tex]
There are two possible values for t, both valids because they are positive
[tex]\displaystyle t=\frac{5}{2}=2.5, \ t=10[/tex]
We need to find the first value, i.e.
[tex]t=2.5 \ hours[/tex]
Now for the graphic method, we plot the graph for the function and a horizontal line at c=4 to find the values of t.
The graph is shown in the image provided below. We can see both values where the funcion and C=4 intersect. Both values coincide with the previously analitically found values
