Question

Find the composite function N(T(t))=Find the time when the bacteria found count reaches 25036

Answer 1

Ok, so

We got the function:

$N(T)=24T^2-143T+85$

Which represents the number of bacteria in function of the temperature of the food.

We also have the function:

$T(t)=8t+1.2$

Where t is the time in hours.

To find the composite function:

$N(T(t))$

We have to evaluate the function N, in T(t).

This is,

$\begin{gathered} N(T(t))=24(8t+1.2)^2-143(8t+1.2)+85 \\ N(T(t))=24(64t^2+2(8t)(1.2)+(1.2)^2)-143(8t)-143(1.2)+85 \\ N(T(t))=24(64t^2+19.2t+1.44)-1144t-171.6+85 \\ N(T(t))=1536t^2+460.8t+34.56-1144t-171.6+85 \\ N(T(t))=1536t^2-683.2t-52.04 \end{gathered}$

Therefore,

N(T(t)) = 1536t^2 - 683.2t - 52.04

We want to find the time when the bacteria count reaches 25036.

For this, we have to find "t" such that N(T(t)) = 25036.

Then, we write:

$\begin{gathered} 25036=1536t^2-683.2t-52.04 \\ 1536t^2-683.2t-52.04-25036=0 \\ 1536t^2-683.2t-25088.04=0 \end{gathered}$

If we solve this quadratic equation, we got that:

$\begin{gathered} 1536t^2-683.2t-25088.04=0 \\ t=4.27 \end{gathered}$

Therefore, the time needed is equal to 4.27 hours