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A cup of coffee at 100 degrees celsius is put into a 25 degree celsius room when t=0. The coffee's temperature, f(t), is changing at a rate given by f′(t)=−9(0.9)t degrees Celsius per minute, where t is in minutes.

Estimate the coffee's temperature when t=9:

f'(t) = -9(0.9)^t

f(t) = -9/ln0.9 (0.9)^t + C
since f(0) = 100, we have C = 100 + 9/ln0.9 ≈ 14.579
so now f(9) = 47.67°

Did you know?

Did you know that the temperature of a cup of coffee can be estimated using the derivative of its temperature function? In this scenario, when a cup of coffee at 100 degrees Celsius is placed in a 25 degrees Celsius room, the coffee's temperature is expected to change over time. The rate of change in temperature, represented by f′(t)=-9(0.9)t degrees Celsius per minute, allows us to estimate the coffee's temperature at different time points. Now, if we want to estimate the temperature after 9 minutes, we can substitute t=9 into the given equation. By doing so, we can calculate f′(9)=-9(0.9)9=-72.9 degrees Celsius per minute. This estimation allows us to predict that the coffee's temperature at t=9 will decrease by approximately 72.9 degrees Celsius, resulting in a temperature of around 27.1 degrees Celsius.