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:

  1. Define the Variables:

    • Let the length of the cable be $X$ meters.
    • The cost of the cable is Rs. 200.
  2. Determine the Cost per Meter:

    • The cost per meter of the cable is given by $$ \text{cost per meter} = \frac{200}{X} $$
  3. 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: $$ (X + 5) \left( \frac{200}{X} - 2 \right) = 200 $$
  4. 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: $$ X^2 - 5X - 500 = 0 $$
  5. 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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