8/11/2014 The number of bacteria, in thousands, in a Petri dish is given by the function: N(T) = 15T2 – 100T + 1000, 5 < T < 20

where T denotes the temperature in Celsius. The temperature of the Petri dish is a function of time, t, in hours: T(t) = 3t + 7, 0 < t < 5

(i) Find the number of bacteria in the Petri dish after 3 hours have elapsed.

(ii) If 1,500,000 bacteria are found in the Petri dish, how many hours must have elapsed?



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8/11/2014 Solution. (i) After 3 hours, the temperature in the Petri dish is:

T(3) = 33 + 7 = 16 Celsius.

The number of bacteria (in thousands), then, is: N(16) = 15162 – 10016 + 1000 = 3240.

(ii) If 1500 (in thousands) bacteria are found in the Petri dish: N(T) = 1500 = 15T2 – 100T + 1000.

Solving the quadratic function for T:

15T2 – 100T – 500 = 0 or 3T2 – 20T – 100 = 0 => T = 10.

Therefore, for the number of hours that must have elapsed: T(t) = 10 = 3t + 7 => t = 1hour.