An electric cable costs Rs. 200. If the cable was 5 meters longer and each meter of cable costs Rs. 2 less, the cost of the cable would have remained unchanged. Find the length of the cable.
Question
An electric cable costs Rs. 200. If the cable was 5 meters longer and each meter of cable costs Rs. 2 less, the cost of the cable would have remained unchanged. Find the length of the cable.
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Answer
To find the length of the electric cable, we can follow these steps:

Define the Variables:
 Let the length of the cable be $X$ meters.
 The cost of the cable is Rs. 200.

Determine the Cost per Meter:
 The cost per meter of the cable is given by $$ \text{cost per meter} = \frac{200}{X} $$

Set Up the Equation:
 If the cable was 5 meters longer and each meter of the cable costs Rs. 2 less, the expression for the cost per meter becomes $$ \left(\frac{200}{X}  2\right) $$
 According to the problem, the total cost would remain Rs. 200, so we set up the equation: $$ (X + 5) \left( \frac{200}{X}  2 \right) = 200 $$

Solve the Equation:
 Simplify and solve for $X$: $$ (X + 5) \left( \frac{200  2X}{X} \right) = 200 $$
 Distribute and combine like terms to form a quadratic equation: $$ X^2  5X  500 = 0 $$

Find the Roots:

Solve this quadratic equation using the quadratic formula, $$ X = \frac{b \pm \sqrt{b^2  4ac}}{2a} $$ where $a = 1$, $b = 5$, and $c = 500$.

On solving the quadratic equation, we obtain: $$ X \approx 23.2548 \text{ meters} $$

Therefore, the length of the cable is approximately 23.2548 meters.
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