Question

The swimming pool at the local YMCA holds roughly 749,511.5 L (749,511.5 kg) of water and is kept at a temperature of 80.6 °F year-round using a natural gas heater. If you were to completely drain the pool and refill it with 50°F water, how many GJ (giga-Joules) of energy are required to heat the water back to 80.6 °F? The specific heat capacity of water is 4182 J/kg⋅°C, and the cost of natural gas per GJ is $2.844. It costs $ (blank) to heat the pool (to the nearest dollar).

a) 71 GJ, $203
b) 120 GJ, $340
c) 82 GJ, $233
d) 94 GJ, $268

Answer 1

To heat the pool from 50°F to 80.6°F, it requires approximately 120 GJ of energy, costing $340.

Therefore, correct answer is b) 120 GJ, $340.

Step-by-step explanation:

The energy required to heat the water can be calculated using the formula:

`\[ Q = m \cdot c \cdot \Delta T \]`

where:

- Q is the heat energy,

- m is the mass of water,

- c is the specific heat capacity of water, and

-
`\( \Delta T \)` is the change in temperature.

Firstly, calculate the mass of water using the density formula:

`\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]`

Rearrange the formula to find the mass:

`\[ \text{Mass} = \text{Density} \cdot \text{Volume} \]`

Given that the density of water is approximately 1 kg/L, the mass of water is 749,511.5 kg.

Next, calculate the heat energy using the specific heat capacity formula:

`\[ Q = 749,511.5 \, \text{kg} \cdot 4182 \, \text{J/kg} \cdot (80.6 - 50) \, \text{°C} \]`

This results in approximately 120 GJ of energy required.

To find the cost, convert GJ to dollars using the cost per GJ:

`\[ \text{Cost} = 120 \, \text{GJ} \cdot \$2.844/\text{GJ} \]`

This gives a total cost of $340.

Therefore, correct answer is b) 120 GJ, $340.