Suppose that the number of bacteria in a plate of food is given by N(T)= 10T^2 - 40T + 200, -1<=T<=20
where T is the temperature of the food in degrees Celsius. Suppose that the temperature of the food when it is left out on a picnic table is given by T(t)= 4t + 10, 0<=t<=2
where t is the time in hours.
a) find the composition of N(T(t)). What is the physical interpretation of this composition function?
I found the composition of the function to be: N(T(t))= 1600t^2 + 7984t + 10160, I just don't understand what the physical interpretation is.
To find the composition of N(T(t)), we need to substitute T(t) into N(T).
Given N(T) = 10T^2 - 40T + 200, we substitute T = T(t) = 4t + 10 into this equation:
N(T(t)) = 10(4t + 10)^2 - 40(4t + 10) + 200
Expanding and simplifying, we get:
N(T(t)) = 10(16t^2 + 80t + 100) - 160t - 400 + 200
N(T(t)) = 160t^2 + 800t + 1000 - 160t - 400 + 200
N(T(t)) = 160t^2 + 640t + 800
So the composition of N(T(t)) is N(T(t)) = 160t^2 + 640t + 800.
Now, let's discuss the physical interpretation of this composition function.
The composition function N(T(t)) gives us the number of bacteria in the plate of food at a specific time t, considering the relationship between the temperature T and time t.
In this case, T(t) = 4t + 10 represents the temperature of the food at time t when it is left out on the picnic table. By substituting this temperature function into the equation for the number of bacteria N(T), we get N(T(t)).
So, the physical interpretation of N(T(t)) is that it represents the number of bacteria in the plate of food at a specific time t, accounting for the temperature of the food at that time.
This composition function helps us understand how the number of bacteria changes over time, depending on the temperature of the food.