A BATH CONTAINS 100 KG OF WATER AT 60 DEGREES CELCIUS. HOT AND COLD TAPS ARE THEN TURNED ON TO DELIVER 20KG PER MINUTE EACH A TEMPERATURE 70 DEGREES CELSIUS AND 10 DEGREES CELSIUS RESPECTIVELY. HOW LONG WILL IT BE BEFORE THE TEMPERATURE IN THE BATH HAS DROPPED TO 45 DEGREES CELCIUS. ASSUME COMPLETE MIXING OF WATER AND IGNORE HEAT LOSSES.

7.5 minutes

Need answers

To find the time it takes for the temperature in the bath to drop to 45 degrees Celsius, we need to calculate the heat energy gained and lost during the process. We can then use this information to determine the time it takes to reach the target temperature.

Let's break down the problem step by step:

Step 1: Calculate the initial heat energy of the bath:
The heat energy (Q) of an object is given by the formula: Q = mcΔθ

Given:
Mass of water in the bath (m) = 100 kg
Initial temperature of the water (θ_1) = 60 °C
Specific heat capacity of water (c) = 4,186 J/kg·°C (approximately)

Using the formula, we can calculate the initial heat energy of the bath:
Q_initial = mcΔθ
= 100 kg * 4,186 J/kg·°C * (60 °C - 45 °C)
= 62,790,000 J

Step 2: Calculate the heat energy gained and lost while mixing:
The heat energy gained or lost during this process is given by the formula: Q = mcΔθ

Given:
Mass of hot water supplied per minute (m_hot) = 20 kg
Temperature of hot water supplied (θ_hot) = 70 °C
Mass of cold water supplied per minute (m_cold) = 20 kg
Temperature of cold water supplied (θ_cold) = 10 °C
Final temperature in the bath (θ_final) = 45 °C

Using the formula, we can calculate the heat energy gained and lost during the process:
Q_hot = m_hot * c * (θ_hot - θ_final)
Q_cold = m_cold * c * (θ_cold - θ_final)

Step 3: Calculate the time it takes to reach the target temperature:
To find the time it takes for the bath to reach the target temperature, we need to consider the rate at which heat energy is gained and lost. Since the heat energy gained is equal to the heat energy lost to reach the target temperature, we can use the formula:

Q_hot + Q_cold = Q_initial

Substituting the known values:
(m_hot * c * (θ_hot - θ_final)) + (m_cold * c * (θ_cold - θ_final)) = Q_initial

Now we solve for time (t):
t = Q_initial / ((m_hot * c * (θ_hot - θ_final)) + (m_cold * c * (θ_cold - θ_final)))

Substituting the known values, we have:
t = 62,790,000 J / ((20 kg * 4,186 J/kg·°C * (70 °C - 45 °C)) + (20 kg * 4,186 J/kg·°C * (10 °C - 45 °C)))

Now, calculate the value of t to find the time it takes to reach the target temperature of 45 °C.