When a cold drink is taken from a refrigerator, its temperature is 5 degrees C. After 25 minutes in a 20 degrees C room its temperature has increased to 10 degrees C. (a) What is the temperature of the drink after 50 minutes? (b) When will its temperature by 15 degrees C?

To solve these questions, we can use Newton's Law of Cooling, which tells us that the rate at which an object changes temperature is proportional to the difference between its own temperature and the ambient temperature.

The formula for Newton's Law of Cooling is given as:

T(t) = T_a + (T_0 - T_a) * e^(-kt)

Where: T(t) = Temperature of the object at time t T_a = Ambient temperature (which is 20 degrees C in this case) T_0 = Initial temperature of the object (5 degrees C when taken out from the refrigerator) k = A constant that depends on the characteristics of the object and its surroundings t = Time in minutes e = base of the natural logarithm

We’re not given the value for the constant k, but we can find it using the information that after 25 minutes, the temperature has increased to 10 degrees C.

10 = 20 + (5 - 20) * e^(-25k)

Let’s solve for k:

-10 = -15 * e^(-25k) 2/3 = e^(-25k) ln(2/3) = -25k * ln(e) ln(2/3) = -25k

k = -ln(2/3) / 25

Now, using this value of k, we can find out the temperature after 50 minutes:

T(50) = 20 + (5 - 20) * e^(-50k)

First, let's calculate e^(-50k):

e^(-50k) = e^(50 * ln(2/3) / 25) = (e^(ln(2/3)))^(50/25) = (2/3)^(50/25) = (2/3)^2 = 4/9

Now put the value in our first equation:

T(50) = 20 + (5 - 20) * (4/9) = 20 + (-15) * (4/9) = 20 - 60/9 = 20 - 6.67 = 13.33 degrees C

So, the temperature of the drink after 50 minutes will be approximately 13.33 degrees C.

To find out when the temperature will be 15 degrees C, we set T(t) to 15 and solve for t:

15 = 20 + (5 - 20) * e^(-kt)

First, let’s solve for e^(-kt):

-5 = -15 * e^(-kt) 1/3 = e^(-kt) ln(1/3) = -kt * ln(e) ln(1/3) = -kt

t = -ln(1/3) / k t = -ln(1/3) / (-ln(2/3) / 25) t = 25 * ln(1/3) / ln(2/3)

Now, calculate the time t:

t ≈ 25 * (-1.0986) / (-0.4055) t ≈ 67.93 minutes

The temperature of the drink will reach approximately 15 degrees C after roughly 67.93 minutes