Question

The number of bacteria in a refrigerated food product is given by N (T) =29T^2 - 63T + 23, 2 < T < 32 where T is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by T (t) = 4t + 1.3 , where t is the time in hours.

Find the complete function N(T(t)):
N(T(t)) =

Find the number of bacteria after 7.5 hours. Give your anwser accurate to the nearest whole value.
__ bacteria

Answer 1

Answer: To find the complete function N(T(t)), we need to substitute T(t) into the equation for N(T) and simplify.

Given:

N(T) = 29T^2 - 63T + 23

T(t) = 4t + 1.3

Substituting T(t) into N(T), we have:

N(T(t)) = 29(T(t))^2 - 63(T(t)) + 23

Now, substitute T(t) with 4t + 1.3:

N(T(t)) = 29((4t + 1.3)^2) - 63(4t + 1.3) + 23

Simplifying this equation step by step:

N(T(t)) = 29(16t^2 + 10.4t + 1.69) - 63(4t + 1.3) + 23

N(T(t)) = 464t^2 + 301.6t + 49.01 - 252t - 81.9 + 23

N(T(t)) = 464t^2 + 49.6t - 8.89

To find the number of bacteria after 7.5 hours, substitute t = 7.5 into N(T(t)):

N(T(t)) = 464(7.5)^2 + 49.6(7.5) - 8.89

Calculating this expression:

N(T(t)) = 26040 + 372 + 23

N(T(t)) = 26435

Therefore, the number of bacteria after 7.5 hours is approximately 26,435 bacteria.

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