When a patient receives a penicillin injection, the kidneys begin removing the penicillin from the body. The table and graph above show the penicillin concentration in a patient’s bloodstream at 5 minute intervals for the 20 minutes immediately following a one-time penicillin injection.
The penicillin concentration, in micrograms per milliliter, in the patient’s bloodstream t minutes after the penicillin injection is modeled by the function P defined by P t = `200b^(t/5)` . If P approximates the values in the table to within 10 micrograms per milliliter, what is the value of b, rounded to the nearest tenth?

Answer

0.8

Explanation

The value of b in the equation `P (t) = 200b^(t/5)` can be estimated using any row of the table other than the first one. Substituting t = 5 and P (5) = 152 from the second row of the table into the definition of P yields `152 = 200b^(5/5)` , or 152 = 200b. Dividing both sides of this equation by 200 yields b = `152/200` . The fraction can be rewritten as `76/100` , or its decimal equivalent .76. Rounded to the nearest tenth, this value is .8. Other rows of the table also give a value of b that rounds to .8. Therefore, the value of b, rounded to the nearest tenth, is .8. Either .8, or its fractional equivalents, 4/5 or 8/10, can be gridded as the correct answer.