Find the time when the bacteria count reaches 23901.

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 :

N(T(t))=

Find the time when the bacteria count reaches 23901.

Time Needed =

Answer 1

In order to find the composite function, remember that N(T(t)) just means we take the whole expression for T(t) and put it everywhere T shows up in N(T). So wherever we see a T, we write 3t + 1.2 instead:

N(T(t)) = 25(3t + 1.2)^2 - 128(3t + 1.2) + 70

Now we can expand it out. Squaring the parentheses first gives us:

(3t + 1.2)^2 = 9t^2 + 7.2t + 1.44

Multiplying that by 25 gives 225t^2 + 180t + 36, and distributing the -128 gives -384t - 153.6. Putting all the pieces together:

N(T(t)) = 225t^2 + 180t + 36 - 384t - 153.6 + 70

N(T(t)) = 225t^2 - 204t - 47.6

For the second part we want to know when the count reaches 23901. You could set that composite equal to 23901 and solve it, but there is an easier way. Since the bacteria count really depends on the temperature, lets first find what temperature gives us 23901 bacteria, and then work backwards to find what time gives us that temperature.

25T^2 - 128T + 70 = 23901

25T^2 - 128T - 23831 = 0

Using the quadratic formula here gives us T = 33.54 degrees. We take the positive root because the other one comes out negative, and the problem tells us the temperature stays between 5 and 35.

Now that we have the temperature, we can put it into T(t) = 3t + 1.2 to find the time:

3t + 1.2 = 33.54

3t = 32.34

t = 10.78

So, our final answer is N(T(t)) = 225t^2 - 204t - 47.6, and the bacteria count reaches 23901 after about 10.78 hours.

Answer 2

1. Find the composite function N(T(t)) by substituting (3t + 1.2) with T in N(T)= 25T^2 − 128T + 70.
2. Let N(T(t)) = 23901, you will get an quadratic equation. Solve the equation to get the time.