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The concentration C (in mg/dl), of an antibiotic in a patient’s bloodstream per hour, t, is given by:
c(t) = (55t)/(t^2+25)

In order for the antibiotic to be effective, 5 or more mg/dl must be present in the bloodstream. When do you have to take the antibiotic after the initial dose? Solve algebraically and graphically.

Respuesta :

LammettHash

We want the concentration of the antibiotic to be at a minimum of 5 mg/dL, so we want to find the time interval, where [tex]t>0[/tex], for which [tex]C(t)\ge5[/tex].


We have


[tex]\dfrac{55t}{t^2+25}\ge5\iff55t\ge5t^2+625\iff t^2-11t+25\le0[/tex]


Completing the square, we get


[tex]t^2-11t+\dfrac{121}4-\dfrac{121}4+25\le0\iff\left(t-\dfrac{11}2\right)^2\le\dfrac{21}4[/tex]


[tex]\implies\left|t-\dfrac{11}2\right|\le\dfrac{\sqrt{21}}2[/tex]


[tex]\implies-\dfrac{\sqrt{21}}2\le t-\dfrac{11}2\le\dfrac{\sqrt{21}}2[/tex]


[tex]\implies\dfrac{11-\sqrt{21}}2\le t\le\dfrac{11+\sqrt{21}}2[/tex]

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