GWCB002 Question 2

Explanation

Part (1)

The amount of electrical energy $W\ \mathrm{J}$W J consumed by an electrical appliance with a power rating of $P\ \mathrm{W}$P W over a time interval of $T\ \mathrm{s}$T s is expressed by the following equation:

$W = P \times T$W=P×T

Substituting the values from the constraints, the required energy $A_1$A1​ can be calculated as follows:

$A_1 = 50 \times 5400 = 2.7 \times 10^5\ \mathrm{J}$A1​=50×5400=2.7×105 J

Subquestion (2)

For the average power consumption in a DC or AC circuit (assuming an ideal state where power is consumed efficiently), the power consumption $P\ \mathrm{W}$P W at a voltage $V\ \mathrm{V}$V V and a current $I\ \mathrm{A}$I A is expressed by the following equation:

$P = I \times V$P=I×V

Substituting the values from the constraints, the average power consumption $A_2$A2​ can be calculated as follows:

$A_2 = 20 \times 200 = 4000\ \mathrm{W}$A2​=20×200=4000 W

Subquestion (3)

The electricity consumption $A_3$A3​ when an air conditioner with the power consumption $A_2 = 4000\ \mathrm{W}$A2​=4000 W (calculated in Subquestion (2)) is used for a time $T_1 = 1.30 \times 10^5\ \mathrm{s}$T1​=1.30×105 s can be calculated as follows using the same equation as in Subquestion (1).

$A_3 = A_2 \times T_1 = 4000 \times (1.30 \times 10^5) = 5.2 \times 10^8\ \mathrm{J}$A3​=A2​×T1​=4000×(1.30×105)=5.2×108 J

Calculation of the Input Form

Substitute the obtained values into the formula for the input form to find the dimensionless ratio.

$\frac{A_3}{A_1 \times A_2} = \frac{5.2 \times 10^8}{(2.7 \times 10^5) \times 4000}$A1​×A2​A3​​=(2.7×105)×40005.2×108​

Calculate the denominator.

$(2.7 \times 10^5) \times 4000 = 1.08 \times 10^9$(2.7×105)×4000=1.08×109

Therefore, the ratio is given by:

$\frac{5.2 \times 10^8}{1.08 \times 10^9} = \frac{5.2}{10.8} = \frac{52}{108} = \frac{13}{27}$1.08×1095.2×108​=10.85.2​=10852​=2713​

Since this is an irreducible fraction, $p = 13$p=13 and $q = 27$q=27. The value we are looking for is $p + q$p+q.

$13 + 27 = 40$13+27=40

The final natural number is 40.