Question

A park is shaped like a rhombus and has area 96 sq m. If 40 m of fencing is needed to enclose the park, the cost, in INR, of laying electric wires along its two diagonals, at the rate of ₹125 per m, is

Answer 1

Correct Answer: 3500

Rhombus Area Calculation:

The area of a rhombus is calculated using the formula:
$ \text{Area} = \dfrac{1}{2} \times d_1 \times d_2 $
where $d_1$ and $d_2$ represent the lengths of the rhombus's diagonals.

Step 1: Utilize Given Area Information

Provided Area = 96.
Therefore,
$96 = \dfrac{1}{2} \times d_1 \times d_2$
This simplifies to:
$d_1 \times d_2 = 96 \times 2 = 192$

Step 2: Apply the Pythagorean Theorem

Each half-diagonal, along with the rhombus side, forms a right-angled triangle.
The given side length of the rhombus is 10 units.

Applying the theorem:
$ \left( \dfrac{d_1}{2} \right)^2 + \left( \dfrac{d_2}{2} \right)^2 = 10^2 $
This expands to:
$ \dfrac{d_1^2}{4} + \dfrac{d_2^2}{4} = 100 $
Multiplying by 4:
$ \dfrac{d_1^2 + d_2^2}{4} = 100 $
Resulting in:
$ d_1^2 + d_2^2 = 400 $

Step 3: Employ the Square of Sum Identity

Utilizing the algebraic identity: $ (a + b)^2 = a^2 + b^2 + 2ab $
For the diagonals, this becomes:
$ (d_1 + d_2)^2 = d_1^2 + d_2^2 + 2d_1d_2 $

Substituting the derived values:
$ (d_1 + d_2)^2 = 400 + 2 \times 192 $
$ = 400 + 384 = 784 $
Taking the square root:
$ d_1 + d_2 = \sqrt{784} = 28 $

Step 4: Calculate the Cost of Electrical Wiring

The cost of laying electric wires is ₹125 per meter.
The total length of wires required, corresponding to the sum of the diagonals, is 28 meters.
Total cost calculation:
Total cost = $28 \times 125 = ₹3500$

Conclusion:

The total expenditure for installing electric wires along the diagonals of the rhombus amounts to ₹3500.