Chapter 22, Problem 99A

(a)

To determine

The amount of energy delivered to the heater in half an hour.

Answer to Problem 99A

The energy drawn by the heater in half an hour is $0.9\times {10}^6\text{J}$ .

Explanation of Solution

Given:

The electric heater is rated at $500\text{W}$ .

The power drawn by the heater is $P=500\text{W}$ .

The time for which the heater operates is $t=0.5\text{h}$

Formula used:

The expression for the energy delivered to the heater is,

$E=Pt$

Here, $P$ is the power used by the heater and $t$ is the time of operation of the motor.

Calculation:

The power drawn by the heater is,

$\begin{array}{l}E=Pt\\ E=500\text{W}\times 0.\text{5}\text{h}\times \left(\frac{3600\text{s}}{1\text{h}}\right)\\ E=0.9\times {10}^6\text{J}\end{array}$

Conclusion:

Thus, the energy drawn by the heater in half an hour is $0.9\times {10}^6\text{J}$ .

(b)

To determine

The change in the air temperature in half an hour.

Answer to Problem 99A

The change in the temperature of the room in half an hour is $8.2°\text{C}$ .

Explanation of Solution

Given:

The mass of air in the room is $m=50\text{kg}$ .

The specific heat of the air is $C=1.10\text{kJ/}\left(\text{kg}\cdot °\text{C}\right)$ .

Consider the change in the temperature of the room is denoted by $\Delta T$ .

The percentage of thermal energy that warms up the room is $50\%$ .

Formula used:

The expression for the change in the temperature $\left(\Delta T\right)$ of the room is,

$\begin{array}{l}H=mC\Delta T\\ \Delta T=\frac{H}{mC}\end{array}$

Here, $m$ is the mass of the air in the room and $C$ is the specific heat of the air.

Calculation:

Refer Part (a).

The energy drawn by the heater in half an hour is $E=0.9\times {10}^6\text{J}$ .

Consider the heat delivered by the heater to the room in half an hour is denoted by $H$ .

The heat delivered by the heater to the room is $50\%$ of heat drawn by the heater.

$\begin{array}{l}H=50\%\times E\\ H=\frac{50}{100}\times 0.9\times {10}^6\text{J}\\ H=4.5\times {10}^5\text{J}\times \left(\frac{\text{1}\text{kJ}}{{10}^3\text{J}}\right)\\ H=4.5\times {10}^2\text{kJ}\end{array}$

The change in the temperature $\left(\Delta T\right)$ of the room in half an hour is,

$\begin{array}{l}\Delta T=\frac{H}{mC}\\ \Delta T=\frac{4.5\times {10}^2\text{kJ}}{\left(50\text{kg}\right)\times 1.10\text{kJ/}\left(\text{kg}\cdot °\text{C}\right)}\\ \Delta T=\frac{4.5\times {10}^5\text{J}}{\left(50\text{kg}\right)\times 1.10\text{kJ/}\left(\text{kg}\cdot °\text{C}\right)}\\ \Delta T=8.18°\text{C}\\ \Delta T\approx 8.2°\text{C}\end{array}$

Conclusion:

Thus, the change in the temperature of the room in half an hour is $8.2°\text{C}$ .

(c)

To determine

The cost of running the heater $6$ hours per day for $30$ days.

Answer to Problem 99A

The cost of running the heater $6$ hours per day for $30$ days is $\$7$ .

Explanation of Solution

Given:

The cost for $1\text{kWh}$ is $\$0.08$ .

The heater runs $6\text{h/day}$ for $30$ days.

Formula used:

Total cost will be the product of cost per kWh and total kWh consumed.

Calculation:

The total duration of operation $\left(t\right)$ of the heater is,

$\begin{array}{l}t=\left(\frac{6\text{h}}{\text{day}}\right)\times 30\text{day}\\ t=180\text{h}\times \left(\frac{3600\text{s}}{1h}\right)\\ t=648,000\text{s}\end{array}$

The heater is rated at $500\text{W}$ . That is, the energy consumed by the heater per second is $500\text{J}$

The energy consumed by the heater in time tis ,

$\begin{array}{l}\text{Energy}\text{Consumed}=\left(\frac{500\text{J}}{\text{s}}\right)\times 648,000\text{s}\\ \text{Energy}\text{Consumed}=324\times {10}^6\text{J}\end{array}$

The cost for $1\text{kWh}$ is $\$0.08$ .

The value of $1\text{kWh}$ is equivalent to $3.6\times {10}^6\text{J}$ .

The cost for $3.6\times {10}^6\text{J}$ is $\$0.08$ .

The cost of running the heater $6$ hours per day for $30$ days is,

$\begin{array}{l}\text{Cost}=\left(\frac{324\times {10}^6\text{J}}{3.6\times {10}^6\text{J}}\right)\times \$0.08\\ \text{Cost}=\$7\end{array}$

Conclusion:

Thus, the cost of running the heater $6$ hours per day for $30$ days is $\$7$ .