Question

The number of bacteria in a refrigerated food product is given by: N(T) = 23T^2 - 56T + 1 for 3 T ; 33, where T is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by T(t) = 5t + 1.5, where t is the time in hours.Find the composite function N(T(t)) = Find the time (round to decimal places) when the bacteria count reaches 6752. t =_____ hours.

Answer 1

To find the composite function N(T(t)), substitute the expression for T(t) into the equation for N(T). Then, solve the quadratic equation for the time when the bacteria count reaches 6752. The time is approximately 6.01 hours.

Step-by-step explanation:

To find the composite function N(T(t)), we need to substitute the expression for T(t) into the equation for N(T).

N(T(t)) = 23(T(t))² - 56(T(t)) + 1

N(T(t)) = 23(5t + 1.5)² - 56(5t + 1.5) + 1

Simplifying the equation, we get:

N(T(t)) = 23(25t² + 15t + 2.25) - 280t - 84 + 1

N(T(t)) = 575t² + 345t + 51.75 - 280t - 83

N(T(t)) = 575t² + 65t - 31.25

We need to find the time t when the bacteria count reaches 6752.

Setting N(T(t)) equal to 6752:

6752 = 575t² + 65t - 31.25

Adding 31.25 to both sides:

6783.25 = 575t² + 65t

This is a quadratic equation. We can solve it using the quadratic formula:

t = (-b ± sqrt(b² - 4ac))/(2a)

Substituting the values into the formula:

t = (-65 ± sqrt(65² - 4(575)(-6783.25)))/(2(575))

Calculating the square root and simplifying, we get:

t ≈ 6.01379

Rounding to two decimal places, the time when the bacteria count reaches 6752 is approximately t = 6.01 hours.