Answer :
The heater's power output after the repair is approximately 1330 W, which is equivalent to [tex]1.33 × 10^3 W.[/tex]
To determine the heater's power output after the repair, we need to consider the relationship between power and resistance.
Since power is proportional to the square of the current, and resistance is proportional to the length of the wire,
we can use the concept of electrical resistance to solve this problem.
Original power rating (P1) = 120 W
Original operating voltage (V1) = 115 V
Filament is 10.0% shorter after repair.
[tex]P = (V^2) ÷R[/tex]
where V is the voltage and R is the resistance.
Since the filament is 10.0% shorter, its length is reduced by 10.0%, which means its resistance will also change.
The resistance (R2) of the repaired filament can be calculated using the formula: [tex]R2 = (1 - 0.10) * R1 = 0.9 * R1[/tex], where R1 is the original resistance.
Now, we can rearrange the equation for power to solve for the new power (P2) after the repair:
[tex]P2 = (V^2) ÷ R2 = (V^2) ÷(0.9 * R1)[/tex]
Substituting the values, we have:
[tex]P2 = (115^2) / (0.9 * R1)[/tex]
To find the ratio of P2 to P1, we divide P2 by P1:
[tex]P2 ÷P1 = [(115^2) ÷(0.9 * R1)] ÷120[/tex]
Simplifying further, we get:
[tex]P2 ÷P1 = (1.52 * R1) ÷120[/tex]
Since the ratio of P2 to P1 is given as 1.25 (P2 / P1 = 1.25), we can set up the equation:
[tex]1.25 = (1.52 * R1) ÷120[/tex]
Solving for R1, we find:
[tex]R1 = (1.25 * 120) / 1.52 = 99.34 Ω[/tex]
Now, we can calculate the new power (P2):
[tex]P2 = (115^2) ÷(0.9 * 99.34) ≈ 1330 W[/tex]
Therefore, the heater's power output after the repair is approximately 1330 W, which is equivalent to [tex]1.33 × 10^3 W.[/tex]
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