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A cup of coffee with cooling constant $k=0.09 \mathrm{~min}^{-1}$ is placed in a room at temperature $20^{\circ} \mathrm{C}$.(a) How fast is the coffee cooling (in degrees per minute) when its temperature is $T=80^{\circ} \mathrm{C} ?$(b) Use the Linear Approximation to estimate the change in temperature over the next 6 s when $T=80^{\circ} \mathrm{C}$.(c) If the coffee is served at $90^{\circ} \mathrm{C}$, how long will it take to reach an optimal drinking temperature of $65^{\circ} \mathrm{C} ?$

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Calculus 2 / BC

Chapter 9

Introduction to Differential Equations

Section 2

Models Involving $y^{\prime}=k(y-b)$

Differential Equations

Campbell University

Oregon State University

University of Michigan - Ann Arbor

Boston College

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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Coffee and his cooling constant is uh .09. And the temperature of the surrounding room is 20°. How fast is it cooling when it's temperatures 80. Use linear approximation to find to find to estimate the change in temperature over the next six seconds. And then if it's served at 90°, how long before it's a drinkable, 60°. Okay, so the first question, how fast is it cooling when its temperature is 80 is asking you What's dy DT when it's 80? So, Dy DT is minus k minus point oh nine time the temperature minus the surrounding temperature, so minus point oh nine times 60 Which is -5.4 degrees per second. So 5°/s is how fast it's cooling. Use linear approximation to estimate the change in temperature over the next six seconds. Okay, so here's what they're saying. Okay, Dy DT is approximately equal to why minus why not? Over x minus X. Not t, sorry. So it's the slope. So then multiply the t minus t not over here. And you get y minus why not equals Dy DT times t minus t nut. So this is the change in temperature here. Dave oddity. And this is, oh, sorry, this is change in temperature of the coffee. And this is the change in time. So Dy DT we know is minus 5.4 And the change in time in six seconds is six seconds. So for -32.4°. All right, now, the next one is what if it's at 90 degrees. So dy DT equals minus point oh nine. Why my that's a weird thing. Y -20. So dy over y minus 20 equals minus 200.9 D. T. Elena Y -20 Equals -1090 Plus C. So why minus 20 equals C. E. To the minus point? Oh 90 Wyatt zero is 90° so 90 -20 equals C. So why equal 70 E. To the minus 700.9 PTI Um plus 20. Forgot to move the 20 over there. Okay so how long will it take to cool down to 60 degrees? So 60 equals 70 E. To the minus 700.9 PTI plus 20. So 40/70 equals each of the -1990 So the Ln of 47 divided by -109 equals T. Uh huh. So the alien of four divided by seven. Good bye bye. Yeah mm point oh nine okay. 6.2 seconds about which is reasonable since that's what it was on. The first two parts of this equation are parts of this question Not 6.2° sorry 6.2 seconds

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