PLEASE HELP! ONLY ANSWER IF YOU KNOW! 100 POINTS 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)?

a

Step-by-step explanation:

  

The only part of the expression representing f(t) involving the variable t is the exponential e−0.08t. Since the coefficient of t in the exponent, −0.08, is negative, the values of f(t) decrease as time passes. Thus the temperature of the coffee decreases as time elapses. This is appropriate as we know from experience that over time the coffee will cool down to room temperature: In this case, the 75 in the expression representing f(t) is the ambient room temperature in degrees Fahrenheit.

The beginning of the experiment is when the time variable t takes the value zero: Since

f(0)=110e−0.08⋅0+75=110+75=185,

the initial temperature of the coffee is 185 degrees Fahrenheit, a little less than the boiling temperature of water.

To find when the coffee is 140 degrees we want to solve

f(t)=110e−0.08t+75=140.

Subtracting 75 from both sides and then dividing both sides by 110 gives

e−0.08t=65110.

By the definition of the natural logarithm, this gives

−0.08t=ln(65110).

Dividing both sides by −0.08 gives a value of about 6.6 minutes for the coffee to cool to 140 degrees.

The same reasoning gives, as an expression for the time when the coffee has cooled to 100 degrees,

ln(25110)−0.08.

This is about 18.5 minutes. sorry if i forgot to put "/" for fractions

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