The differential equation below models the temperature of an 89°C cup of coffee in a 21°C room, where it is known that the coffee cools at a rate of 1°C per minute when its temperature is 71°C. Solve the differential equation to find an expression for the temperature of the coffee at time t. (Let y be the temperature of the cup of coffee in °C, and let t be the time in minutes, with t = 0 corresponding to the time when the temperature was 89°C.) dy dt = − 1 50 (y − 21)

Step-by-step explanation:

Consider the differential equation . NOte that this fullfills the condition that the coffee cools down 1°C when y=71°C. We have the initial condition that y(0) = 89°C. Note that this is a separable differential equation, since

which leads to

. This gives us the following result

where C is a constant. Then, using the exponential function we have

which is another constant.

Given the initial condition, we have that when t=0, y = 89, then

Then, the final solution is

part a

x     y

0     8

1     7.5

2     7

3     6.5

4     6

5     5.5  

for every hour i deducted .5 inches from the length of the candle

part b

it is a function because for every x value there is only one y value

part   c

for every x value there is only one y value

x       y

0       8

1       7.6

2       7.2

3       6.8

step-by-step explanation: