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.
The composition function N(T(t)) = 1600t^2 + 7984t + 10160 represents the number of bacteria in the plate of food as a function of time.
The physical interpretation of this composition function is that it provides a way to determine the number of bacteria in the plate of food at any given time (t) based on the temperature of the food (T). By substituting the temperature function T(t) = 4t + 10 into the expression for N(T), you can calculate the number of bacteria at each point in time.
This composition function allows you to understand how the temperature of the food, which changes over time, influences the growth or decay of the bacteria. It helps in studying the relationship between temperature and bacteria growth in the context of this specific function.
To find the composition of N(T(t)), we need to substitute the expression for T(t) into N(T).
Given N(T) = 10T^2 - 40T + 200 and T(t) = 4t + 10, we can substitute T(t) into N(T):
N(T(t)) = 10(T(t))^2 - 40(T(t)) + 200
Now replace T(t) with its value:
N(T(t)) = 10(4t + 10)^2 - 40(4t + 10) + 200
Expanding and simplifying:
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)) represents the number of bacteria in the plate of food (N) at a given time (t), based on the temperature of the food (T).
The equation N(T(t)) = 160t^2 + 640t + 800 shows that the number of bacteria varies with time. The coefficient of t^2 (160) shows that the number of bacteria increases or decreases quadratically with time. The coefficient of t (640) indicates a linear change in the number of bacteria with time. The constant term (800) represents a baseline or initial number of bacteria.
Overall, the physical interpretation of N(T(t)) is that it gives the number of bacteria in the plate of food at a certain time, taking into account the temperature of the food at that time.