SOLUTION: The number of bacteria in a refrigerated food product is given by N(T)=24T2-108T+68, 4<T<34 where T is the temperature of the food. When the food is removed from the refrig

Question 1187399: The number of bacteria in a refrigerated food product is given by
N(T)=24T2-108T+68, 4 When the food is removed from the refrigerator, the temperature is given by
T(t)=9t+1.3, where t is the time in hours.
Find the composite function
N(T(t))=1944t2−410.4t−31.84. Correct

Find the number of bacteria after 3.2 hours. Give your answer accurate to the nearest whole value.
? bacteria
The first part I got it correct I only need help with the secont part pleas.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
here's how it works, i believe.

you are given that t = 3.2.

solve for T(t) to get:

T(t) = 9t + 1.3 = 9*3.2 + 1.3 = 30.1

that's the value of T.

go back to your original equation and replace T with 30.1.

you will get:

N(T)=24T2-108T+68 becomes:

N(T) = 24 * 30.1^2 - 108 * 30.1 + 68 = 18561.44.

now go back to your composite function and replace t with 3.2 to get:

N(T(t))=1944t2−410.4t−31.84 becomes:

N(T(t)) = 1944 * 3.2^2 - 410.4 * 3.2 1 31.84 = 18561.44

the answer is exactly the same.

it should be right on because the composite function is the same as solving for T(t) first and then using the result of that as input to N(T).

that's what composite functions do.

they combine the operations of 2 equations into 1 equation.
you used T(t) to get T.
you then place T in the equation of N(T) to get N.

i'll give you a simpler example.

f(x) = x^2
g(x) = (x-1)
f(g(x)) = (x-1)^2 = x^2 - 2x + 1

if x = 5, then f(5) = 5^2 = 25 and g(x) = 4.

f(g(x)) = (5-1)^2 = 4^2 = 16

when you solve g(x) first, you get x = 4.
you then solve for f(g(x)) = f(4) = 4^2 = 16.

the composite function is the same as first solving for the inner function and then using the value from the inner function to solve for the outer function.

T(t) was the inner function.
N(T) was the outer function.

here's a reference on composite functions.

https://www.mathsisfun.com/sets/functions-composition.html

let me know if you have any questions.

theo

good job on finding the composite function, by the way.

fyi, i messed up the original calculations because i wrote the equation down wrong.
i have since corrected it.
this is the corrected version.

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