Math Snap

STEP 1

Assumptions1. The mass of water to be heated is190 kg. The initial temperature of the water is27°C3. The final temperature of the water is100°C4. The specific heat of water is4.184 J/g°C5. The conversion factor from pounds to grams is1 lb =454 g6. The energy content of oil is45 MJ/kg7. The energy required to heat the water is provided entirely by the oil

STEP 2

First, we need to calculate the amount of energy required to heat the water. This can be done using the formula$$Q = m \cdot c \cdot \Delta$$where- Q is the heat energy required (in Joules)
- m is the mass of the water (in grams)
- c is the specific heat of water (in J/g°C)
- Δ is the change in temperature (in °C)

STEP 3

Convert the mass of the water from kilograms to grams.
$$m =190 \, kg =190,000 \, g$$

STEP 4

Calculate the change in temperature.
$$\Delta =_{final} -_{initial} =100°C -27°C =73°C$$

STEP 5

Now, plug in the values for the mass of the water, the specific heat of water, and the change in temperature to calculate the heat energy required.
$$Q =190,000 \, g \cdot4.184 \, J/g°C \cdot73°C$$

STEP 6

Calculate the heat energy required.
$$Q =190,000 \, g \cdot4.184 \, J/g°C \cdot73°C =5.8 \times10^9 \, J$$

STEP 7

Convert the heat energy required from Joules to MegaJoules.
$$Q =5. \times10^9 \, J =5. \times10^3 \, MJ$$

STEP 8

Now, we can calculate the mass of oil required to provide this amount of energy. This can be done using the formula$$m_{oil} = \frac{Q}{_{oil}}$$where- m_oil is the mass of oil required (in kg)
- Q is the heat energy required (in MJ)
-_oil is the energy content of oil (in MJ/kg)

STEP 9

Plug in the values for the heat energy required and the energy content of oil to calculate the mass of oil required.
$$m_{oil} = \frac{5.8 \times^3 \, MJ}{45 \, MJ/kg}$$

Calculate the mass of oil required.
$$m_{oil} = \frac{5.8 \times10^3 \, MJ}{45 \, MJ/kg} =129 \, kg$$To two significant figures, the mass of oil required is130 kg.