Mr. Currie pours himself some coffee into a paper cup before making his way to the amusement park. The coffee temperature is 350C when the cup is placed on the kitchen counter with room temperature of 20o C. Mr. Currie was called to the phone for last minute arrangements, and his coffee is forgotten. When he finally returns to his coffee 35 minutes later, the temperature had been decreasing by 1.2% per minute.

a) What type of function best models the cooling of a hot liquid?
Explain your choice.

b) What is the mathematical model for this situation? (i.e. - the equation)

c) If the optimal temperature for drinking a hot liquid is 280C, at what time
would Mr. Currie have had to return in order to enjoy his cup of coffee?

a) The cooling of a hot liquid can be modeled by an exponential decay function. This is because as time passes, the temperature of the liquid decreases at a decreasing rate. Exponential decay functions are commonly used to model processes that have a constant percentage decrease over time.

b) The mathematical model for this situation can be represented by the equation: T(t) = T₀ * (1 - r)^t, where T(t) is the temperature of the coffee at time t, T₀ is the initial temperature of the coffee, r is the rate of temperature decrease per minute (expressed as a decimal), and t is the time in minutes.

c) To find the time at which the coffee reaches the optimal temperature of 28 degrees Celsius, we can plug in the known values into the equation:

280 = 350 * (1 - 0.012)^t

Now we need to solve for t.

a) The cooling of a hot liquid can be modeled by an exponential decay function. This is because the rate of temperature decrease is proportional to the current temperature of the liquid.

b) The mathematical model for this situation can be represented by the equation: T(t) = T0 * e^(-rt), where T(t) is the temperature of the liquid at time t, T0 is the initial temperature of the liquid, e is the mathematical constant approximately equal to 2.71828, r is the rate of temperature decrease (in this case, 1.2% per minute), and t is the time in minutes.

c) To find the time at which Mr. Currie would have had to return in order to enjoy his cup of coffee at the optimal temperature of 28°C, we need to solve the equation T(t) = 280. Substituting the given values into the equation, we have:

280 = 350 * e^(-0.012t)

To solve for t, we can take the natural logarithm (ln) of both sides:

ln(280) = ln(350 * e^(-0.012t))

Using the property of logarithms, we can simplify this to:

ln(280) = ln(350) + (-0.012t)

Rearranging the equation to solve for t:

-0.012t = ln(280) - ln(350)

Dividing both sides by -0.012:

t = (ln(280) - ln(350)) / -0.012

Evaluating this expression using a calculator, we can find the value of t, which represents the time in minutes that Mr. Currie would have had to return to enjoy his coffee at the optimal temperature of 280°C.