A body cools from `70^(@)C" to "60^(@)C` in 5 minutes and to `45^(@)C` in the next 10 minutes. Calculate the temperature of the surroundings.

Answer

To solve the problem of finding the temperature of the surroundings using Newton's law of cooling, we will follow these steps:

### Step 1: Understand the Problem
We need to find the temperature of the surroundings (θ₀) given the cooling of a body from 70°C to 60°C in 5 minutes and then from 60°C to 45°C in the next 10 minutes.

### Step 2: Use Newton's Law of Cooling
Newton's law of cooling states that the rate of change of temperature of an object is proportional to the difference between its own temperature and the ambient temperature (temperature of surroundings). The formula is given by:

\[ \frac{d\theta}{dt} = -k(\theta - \theta_0) \]

Where:
- θ is the temperature of the body,
- θ₀ is the temperature of the surroundings,
- k is a constant.

### Step 3: Calculate Average Temperatures
1. For the first interval (70°C to 60°C):
- Average temperature (θ₁) = (70 + 60) / 2 = 65°C

2. For the second interval (60°C to 45°C):
- Average temperature (θ₂) = (60 + 45) / 2 = 52.5°C

### Step 4: Set Up the Equations
We can set up two equations based on the cooling process:

1. From 70°C to 60°C in 5 minutes:
\[ \frac{70 - 60}{5} = -k(65 - \theta_0) \]
Simplifying gives:
\[ 2 = -k(65 - \theta_0) \quad \text{(Equation 1)} \]

2. From 60°C to 45°C in 10 minutes:
\[ \frac{60 - 45}{10} = -k(52.5 - \theta_0) \]
Simplifying gives:
\[ 1.5 = -k(52.5 - \theta_0) \quad \text{(Equation 2)} \]

### Step 5: Solve the Equations
Now we will solve these two equations simultaneously.

From Equation 1:
\[ k = \frac{-2}{65 - \theta_0} \]

From Equation 2:
\[ k = \frac{-1.5}{52.5 - \theta_0} \]

Setting the two expressions for k equal to each other:
\[ \frac{-2}{65 - \theta_0} = \frac{-1.5}{52.5 - \theta_0} \]

Cross-multiplying gives:
\[ -2(52.5 - \theta_0) = -1.5(65 - \theta_0) \]

Expanding both sides:
\[ -105 + 2\theta_0 = -97.5 + 1.5\theta_0 \]

Rearranging terms:
\[ 2\theta_0 - 1.5\theta_0 = 105 - 97.5 \]
\[ 0.5\theta_0 = 7.5 \]
\[ \theta_0 = 15°C \]

### Final Answer
The temperature of the surroundings (θ₀) is **15°C**.

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