**8/11/2014
**The
number of bacteria, in thousands, in a Petri dish is given by the
function: N(T) = 15T^{2}
– 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*) = 3*t *+
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?

**Scroll
down for the solution.**

**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) = 1516^{2}
– 10016 + 1000 =
3240.

(ii) If 1500 (in thousands) bacteria
are found in the Petri dish: N(T) = 1500 = 15T^{2} –
100T + 1000.

Solving the quadratic function for T:

15T^{2} – 100T –
500 = 0 or 3T^{2} – 20T – 100 = 0 => T = 10.

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