Use Newton's Law of Cooling. A hot bowl of soup is served at...

Use Newton's Law of Cooling. A hot bowl of soup is served at a dinner party. It starts to cool according to Newton's Law of Cooling, so its temperature at time $t$ is given by $$ T(t)=65+145 e^{-0.05 t} $$ where $t$ is measured in minutes and $t$ is measured in "F. a. What is the initial temperature of the soup? b. What is the temperature after $10 \mathrm{min}$ ? c. After how long will the temperature be $100^{\circ} \mathrm{F}$ ?

Use Newton's Law of Cooling.
A hot bowl of soup is served at a dinner party. It starts to cool according to Newton's Law of Cooling, so its temperature at time $t$ is given by
$$ T(t)=65+145 e^{-0.05 t} $$
where $t$ is measured in minutes and $t$ is measured in "F.
a. What is the initial temperature of the soup?
b. What is the temperature after $10 \mathrm{min}$ ?
c. After how long will the temperature be $100^{\circ} \mathrm{F}$ ?

Key Concepts

Solving Exponential Equations

This concept involves techniques for handling equations in which the variable of interest appears in an exponent. It often requires taking logarithms to isolate the variable. This procedure is key when determining the time required for a quantity undergoing exponential decay (or growth) to reach a particular value.

Initial Conditions

Initial conditions are the starting values provided for a function, such as the initial temperature of an object. These conditions are essential for accurately modeling the behavior of systems described by differential equations, as they allow one to determine the specific solution that fits the physical scenario.

Exponential Decay

Exponential decay is a process in which the rate of decrease of a quantity is directly proportional to its current value. In the context of cooling, it captures how the difference between an object's temperature and the ambient temperature decreases over time, described using an exponential function.

Newton's Law of Cooling

This principle governs the rate at which an object changes temperature through heat transfer, stating that the rate of change of the temperature of an object is proportional to the difference between its current temperature and the ambient temperature. It is commonly modeled by a differential equation and describes a cooling (or heating) process that often results in an exponential function.

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