Question

Newton's Law of Cooling The temperature, F(t), of a heated object at time t is given by F(t) = C + (To - C)e^kt,

where C is the constant temperature of the surrounding medium, To is the initial temperature of the heated object, and k is a negative cooling constant that is associated with the object.
(a) Suppose a freshly brewed cup of coffee is 170° F and sits in a room with a constant temperature of 68°. After 30 minutes the temperature of the coffee is 90°. Use Newton'sLaw of Cooling to find a model for the temperature of the coffee, F(t), after t minutes.

Answer 1

To create a model for the coffee's temperature after t minutes using Newton's Law of Cooling, we use the given initial conditions and solve for the cooling constant k. Then we can express F(t) completely to predict future temperatures.

Step-by-step explanation:

To find the model for the temperature of the coffee, F(t), after t minutes using Newton's Law of Cooling, we need to use the formula F(t) = C + (To - C)eat, where:

• C is the constant temperature of the surrounding medium (68° F),
• To is the initial temperature of the object (170° F), and
• k is the cooling constant we need to find.

Given that after 30 minutes the temperature of the coffee is 90° F, we can substitute the values into the equation to find the value of k:

90 = 68 + (170 - 68)e30k

Solving for k, we find the cooling constant. Once we have k, we can write the complete model for the temperature of the coffee as a function of time, t, in minutes.