# A freshly brewed cup of coffee has temperature 95 degrees C in a 20 degree C room. When its...

## Question:

A freshly brewed cup of coffee has temperature 95{eq}^{\circ} {/eq}C in a 20{eq}^{\circ} {/eq}C room. When its temperature is 63{eq}^{\circ} {/eq}C, it is cooling at a rate of 1{eq}^{\circ} {/eq}C per minute.

When does this occur (in minutes)? (Round answer to two decimal places.)

## Newton's law of cooling

From Newton's law of cooling, it can be said that the heat loss rate is directly proportional to the difference in temperature of a given body and the surrounding in which it is placed.

The equation for temperature of coffee using Newton's law of cooling :

{eq}T(t)=20+Ce^{kt} {/eq}

For initial time at t=0, the equation becomes

{eq}\\ 95=T(0)=20+Ce^{k\cdot 0}=20+C \\ C=75 {/eq}

So, the equation becomes,

{eq}T(t)=20+75e^{kt} {/eq}

After some time at {eq}t=t_o {/eq},

{eq}63=T(t_0)=20+75^{kt_0} \\75e^{kt_0}=43 \\e^{kt_0}=\frac{43}{75} = 0.573 \\kt_0=ln\ (\frac{43}{75})= -0.556 {/eq}

The rate of cooling is at a rate of 1{eq}^{\circ} {/eq}C per minute.

{eq}\\-1=T'(t_o)=75(e^{kt_o} k)=75ke^{kt_o} \\k=\frac{-1}{75e^{kt_o} }=\frac{-1}{75(0.573)}=-0.023 {/eq}

{eq}to=\frac{ln(\frac{43}{75})}{k}={\frac{-0.556}{-0.023}}=24.17\ minutes {/eq}

The temperature of coffee is 63{eq}^{\circ} {/eq}C after 24.17 minutes.

Types of Heating & Cooling Systems

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Chapter 10 / Lesson 9
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You can choose from a number of different heating and cooling systems to make your home comfortable to live in. Watch this video to learn about the advantages and disadvantages of forced air, radiant heat, baseboard heat and geothermal systems.