Question

The number of bacteria in a refrigerated food is given by the function N(T) = 5T^2 – 10T + 100, where T is the temperature of the food in degrees Celsius. To find the temperature at which the number of bacteria is minimal, we need to determine the minimum of this function.

Calculate the derivative of N(T) with respect to T:
N'(T) = d/dT (5T^2 – 10T + 100)

Answer 1

The derivative of N(T) = 5T² – 10T + 100 is N'(T) = 10T - 10, which can be used to find the temperature at which bacteria growth is minimized in food.

Step-by-step explanation:

The derivative of the function N(T) = 5T2 – 10T + 100 with respect to temperature T is found by applying the power rule of differentiation. Calculate N'(T) by taking the derivative of each term with respect to T:

• Derivative of 5T2 is 10T,
• Derivative of – 10T is – 10,
• The derivative of a constant (+100) is 0.

Combining these, we get N'(T) = 10T - 10. Setting this derivative to zero and solving for T will give the temperature at which the number of bacteria is minimal, which corresponds to the vertex of the parabola on a graph representing the function.

Understanding the function's implications in relation to bacterial growth is vital for food safety. Bacteria multiply rapidly at certain temperatures, and controlling the temperature can significantly affect their growth rate. This function might describe the changes in the number of bacteria at varying temperatures within a refrigerator, where failure of temperature control can lead to health risks.