Understanding Composition of Functions and Bacteria Growth

This video tutorial explains the application of composition of functions in determining the number of bacteria in a refrigerated food product. It introduces two functions: one for bacteria count based on temperature and another for temperature change over time. The tutorial guides through finding the composition function, simplifying it, and using the quadratic formula to determine the time when the bacteria count reaches 6,000. The process involves replacing variables, distributing terms, and solving quadratic equations.

What is the initial function that describes the number of bacteria in the food product?

• n(T) = 5T + 1.3

• n(T) = 364T + 47.32

• n(T) = 25T^2 - 78T + 4

• n(T) = 700T^2 - 26T - 50.8

How is the temperature of the food related to time after removal from the refrigerator?

• T(t) = 5t + 1.3

• T(t) = 25t^2 - 78t + 4

• T(t) = 364t + 47.32

• T(t) = 700t^2 - 26t - 50.8

What is the first step in finding the composition function n(T(t))?

• Distribute constants

• Combine like terms

• Replace T with 5t + 1.3 in the function n

• Solve the quadratic equation

What is the result of simplifying the expression 5t + 1.3 squared?

• 5t^2 + 1.3t + 1.69

• 25t^2 + 13t + 1.69

• 5t^2 + 13t + 1.3

• 25t^2 + 10t + 1.3

What is the final form of the function n as a function of time t?

• n(t) = 364t + 47.32

• n(t) = 25t^2 - 78t + 4

• n(t) = 700t^2 - 26t - 50.8

• n(t) = 5t + 1.3

What equation do we solve to find the time when the bacteria count reaches 6,000?

• 364t + 47.32 = 6000

• 700t^2 - 26t - 50.8 = 6000

• 25t^2 - 78t + 4 = 6000

• 5t + 1.3 = 6000

Which method is used to solve the equation for the time when the bacteria count reaches 6,000?

• Elimination method

• Graphical method

• Substitution method

• Quadratic formula

What are the values of a, b, and c in the quadratic equation used to find the time?

• a = 700, b = -26, c = -5950.8

• a = 25, b = -78, c = 4

• a = 5, b = 1.3, c = 0

• a = 364, b = 47.32, c = 0

What is the approximate time it takes for the bacteria count to reach 6,000?

• 2.9585 hours

• 3.5 hours

• 4.2 hours

• 1.5 hours

Why is the negative solution of the quadratic equation not considered?

• It does not satisfy the original equation

• Time cannot be negative

• It results in a negative bacteria count

• It is not a real number