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Step-by-Step Solution
Step 1: Identify Key Information
• Coil A boils water in 10 minutes.
• Coil B boils water in 40 minutes.
• Both coils are now connected in parallel,
• We need to find the new boiling time, $t_p$.
Step 2: Express the Heat Required
Let $H$ be the amount of heat required to boil the water. This heat is delivered by the power dissipated in the resistor over the boiling time. If $R_1$ is the resistance of coil A and $R_2$ is the resistance of coil B, and $V$ is the supply voltage, then:
$$
H = \text{Power} \times \text{Time}
$$
For coil A:
$$
H = \frac{V^2}{R_1} \times t_1
$$
where $t_1 = 10 \text{ minutes}
$$
\\
For coil B:
$$
H = \frac{V^2}{R_2} \times t_2
$$
where $t_2 = 40 \text{ minutes}
$$
Step 3: Relate the Resistances Using Times
Because each coil alone produces the same heat $H$, we have:
$$
\frac{V^2 \, t_1}{R_1} = \frac{V^2 \, t_2}{R_2}
$$
Simplifying, we get:
$$
\frac{t_1}{R_1} = \frac{t_2}{R_2}
$$
Hence,
$$
R_1 : R_2 = t_1 : t_2 = 10 : 40 = 1 : 4
$$
Step 4: Calculate Effective Resistance in Parallel
When the two coils are connected in parallel, their effective resistance $R_\text{p}$ is:
$$
R_\text{p} = \frac{R_1\, R_2}{R_1 + R_2}
$$
Step 5: Power in Parallel and Required Time
The same amount of heat $H$ is required to boil the water, but now the power is supplied by the parallel combination:
$$
H = \frac{V^2}{R_\text{p}} \times t_p
$$
Using the relationship $H = \frac{V^2}{R_1} \times t_1$, we get:
$$
\frac{V^2}{R_1} \times t_1 = \frac{V^2}{R_\text{p}} \times t_p
$$
This leads to:
$$
t_p = t_1 \times \frac{R_\text{p}}{R_1}
$$
Alternatively (as often done in such problems), one can directly use the result for paralleling heating elements:
$$
\frac{1}{t_p} = \frac{1}{t_1} + \frac{1}{t_2}
$$
Both approaches will yield the same final expression.
Step 6: Substitute Values to Find $t_p$
From the formula
$$
\frac{1}{t_p} = \frac{1}{t_1} + \frac{1}{t_2},
$$
we get:
$$
\frac{1}{t_p} = \frac{1}{10} + \frac{1}{40} = \frac{4}{40} + \frac{1}{40} = \frac{5}{40} = \frac{1}{8}.
$$
Hence,
$$
t_p = 8 \text{ minutes.}
$$
Step 7: Final Answer
The time taken by the kettle to boil the same quantity of water when both coils are connected in parallel is 8 minutes.
