A freshly brewed cup of coffee has temperature 95°C in a 20°C room. When its temperature is 68°C, it is cooling at a rate of 1°C per minute. When does this occur? (Round your answer to two decimal places.)

Ah, the race against time! When your freshly brewed cup of coffee is cooling at 1°C per minute, it means it's losing 1°C of heat every minute. So, let's do some math while keeping it light and frothy.

To find out when your coffee reaches 68°C, we have to determine how much heat it loses. From 95°C to 68°C, there's a difference of 27°C. Since it's losing heat at a rate of 1°C per minute, it will take 27 minutes for your coffee to reach 68°C.

Therefore, without any distractions or detours, your coffee will hit the cozy temperature of 68°C after 27 minutes of cooling. Cheers to patience and cool cups of caffeine!

To find out when the coffee reaches a temperature of 68°C, we can use the formula for cooling:

T(t) = T₀ + (T₁ - T₀)e^(-kt),

where T(t) is the temperature at time t, T₀ is the initial temperature, T₁ is the temperature after a certain time, and k is the cooling constant.

Given information:
- Initial temperature (T₀) = 95°C
- Room temperature (T₁) = 20°C
- Cooling rate = 1°C per minute

Substituting the values into the equation, we have:

68 = 95 + (20 - 95)e^(-k * t)

We need to solve for t, the time taken to reach a temperature of 68°C.

Rearranging the equation gives:

(68 - 95) / (20 - 95) = e^(-k * t)

Simplifying further:

-27 / (-75) = e^(-k * t)

27 / 75 = e^(-k * t)

0.36 = e^(-k*t)

To find the time (t) when the coffee reaches 68°C, we need to take the natural logarithm (ln) of both sides:

ln(0.36) = ln(e^(-k * t))

Using the property of logarithms where ln(e^x) = x:

ln(0.36) = -k * t

Now, we know that the coffee is cooling at a rate of 1°C per minute, so k = 1. Plugging in the values:

ln(0.36) = -t

Using a calculator, we find:

t ≈ 1.0177 minutes

Rounding the answer to two decimal places:

t ≈ 1.02 minutes

Therefore, the coffee reaches a temperature of 68°C approximately 1.02 minutes after it starts cooling.

To find the time when the coffee reaches a temperature of 68°C, we can use the concept of cooling rates.

Let's consider the difference in temperature between the coffee and the room at any given time. This temperature difference, also known as the temperature gradient, determines how fast the coffee is cooling.

At the beginning, the temperature difference is (95°C - 20°C) = 75°C. We know that the coffee cools down at a rate of 1°C per minute when its temperature is 68°C.

To find the time it takes for the coffee to cool down from 95°C to 68°C, we can calculate the cooling time for each degree Celsius (°C) decrease in temperature.

The cooling time for the coffee to drop by one degree Celsius is 1 minute when its temperature reaches 68°C. Therefore, the total cooling time required is equal to the temperature difference (75°C) multiplied by one minute per degree Celsius, resulting in 75 minutes.

Therefore, the coffee will reach a temperature of 68°C after 75 minutes.