Composition Functions

Question

The number of bacteria in a refrigerated food product is given by , , where  is the temperature of the food.When the food is removed from the refrigerator, the temperature is given by , where  is the time in hours.

Find the composite function :   

Find the time when the bacteria count reaches 907.Time Needed =    hours

Answer 1

For the composition we plug T(t) into your N(T) function for all instances of T. Looks something like

N = 29T² - 121T + 18

= 29(6t + 1.9)² - 121(6t + 1.9) + 18

= 29 (36t² + 22.8t + 3.61) - 121(6t + 1.9) + 18

= [1,044t² + 661.2t + 104.69] - [726t + 229.9] + 18

= 1,044t² - 64.8t + 352.59

Now to solve for when N(t) equals 907 we set that quadratic equation equal to 907 and solve for t.

907 = 1,044t² - 64.8t + 352.59

Subtract 907 and use the quadratic formula.

0 = 1,044t² - 64.8t - 554.41

The parameters are:

a = 1,044

b = -64.8

c = -554.41

The quadratic formula tells us:

t = (-b ± √(b² - 4ac)) / 2a

The two solutions end up being

t = -0.698 (makes no sense in this context) and t = 0.760

So your answer is t = 0.760 hours.