Question

It's a gross thought, but the number (N) of bacteria in refrigerated food is given by latex-  1≤T≤20 where T is the temperature of the food in degrees Celsius. When you take the food out of the refrigerator, the temperature of the food is given by T(t)=3t+2,   0≤t≤6 where t is the time in hours. Find the composition N(T(t)) and interpret what it means in this context.

Answer 1

Given that the concentration of bacteria in the refrigerated food is

`10T^2-20T-6----\mleft\lbrace1\mright\rbrace`

and the temperature of the food is given by

`T(t)=3t+2-----\mleft\lbrace2\mright\rbrace`

Therefore, N(T(t) is given by

`N\mleft(T(t)\mright)=10(3t+2)^2-20(3t+2)^{}-6`

Then,

`\begin{gathered} N(T(t))=10(3t+2)^2-20(3t+2)^{}-6 \\ =10(9t^2+6t+6t+4)-60t-40-6 \\ =10(9t^2+12t+4)-60t-46 \\ =90t^2+120t+40-60t-46 \\ =90t^2+60t-6 \end{gathered}`

Answer: The composition is

`N(T(t))=90t^2+60t-6`

It can be interpreted as the concentration of bacteria in the food when outside of the refrigerator with time.