An infectious strain of bacteria increases in number at a relative growth rate o
ID: 3116517 • Letter: A
Question
An infectious strain of bacteria increases in number at a relative growth rate of 210 percent per hour. (The relative growth rate is, when written as a decimal, the value of in the formula .) When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 24 hours. How long will it take for the critical level to be reached if the same person is infected with 10 bacteria?Your answer is hours.
Explanation / Answer
We have P(t) = Aert, where A is the initial number of bacteria, P(t) is the number of bacteria after t hours and r = 210/100 = 2.10 is the growth rate of bacteria per hour. Now, when A = 1, we have P(24) = 1*e2.1*24 = e50.4 . This is the critical number of bacteria to make a a person ill. When A = 10, we have P(t) = 10*e2.1t. If P(t) is the critical number of bacteria, then 10*e2.1t = e50.4 so that (e50.4)/( e2.1t) =10 or, e50.4-2.1t = 10. Now, on taking natural logarithm of both the sides, we get ln e50.4-2.1t = ln 10 or, (50.4-2.1t)ln e = ln 10 or, 50.4-2.1t = ln 10 or, 2.1t = 50.4-ln 10 = 50.4- 2.302585093 = 48.09741491. Hence t = 48.09741491/2.1 = 22.90353091 or, 22hours, 54 minutes approximately or, 23 hours ( on rounding off to the nearest hour). The answer is 23 hours.