Question

A cup of hot coffee will, over time, cool down to room temperature. The principle of physics governing the process is Newton's Law of Cooling. The temperature (in degrees Fahrenheit) of a cup of coffee can be modeled by the following equation: f(t)=80+120e^(-0.08t) Here the time t is measured in minutes after the coffee was poured into the cup. A. Explain, using the structure of the expression 80+120e^(-0.08t), why the coffee temperature decreases as time elapses (8 points). B. What is the temperature of the coffee at the beginning of the experiment (7 points)? C. Drinking coffee at 175 degrees can possibly burn your tongue. After waiting for 2 minutes, Leo became impatient and decided to drink the coffee. Will he be alright (7 points)? D. The ideal temperature to drink coffee is between 120 to 140 degrees, as many of the more subtle flavors of coffee are revealed. When is the best time for Leo to drink coffee (8 points)?

Answer 1

A. Because the coefficient of time is negative .

The value of t decreases as time passes .

The temperature of the coffee decreases as time passes and the coffee will evemtually cool down to room temperature

B. At the beginning , t=0

Hence Temperature will be 80 + 120 = 200º

c. He wont be alright . High temperature , less time to cool down . So he wont be alright . Coffee might still be hot .

D. Assuming the temp is 130º

80 + 120ᵅ⁻⁰·⁰⁰⁸ˣ = 130

120ᵅ⁻⁰·⁰⁰⁸ˣ =130-80

120ᵅ⁻⁰·⁰⁰⁸ˣ =50

a-0.008x =(50/120)

0.008x = ln(0.416)

x = 10.962 =11mins

so he has to wait 11 mins to drink.

Answer 2

Explanation:

f(t)=80+120e^(-0.08t)

-----------------------

• A) Temperature decreases over time as the coefficient of time is negative (-0.08)

• B. t=0 ⇒ f(0)= 80 + 120*1 = 200 deg

• C. t=2 ⇒ f(2)= 80+120*e^(-0.08*2) ≈ 182 deg

Coffee is still hot. He won't be alright. He needs to wait more

• D. f(x)= 120

• 80+120e^(-0.08x)= 120
• 120e^(-0.08x)=40
• e^(-0.08x)=1/3
• -0.08x= ln(1/3)
• -0.08x= -1.0986
• x= -1.0986/-0.08
• x ≈ 14 min

best time is ≈ 14 min