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

Area of the Rhombus: 

The area of a rhombus is given by:
$ \text{Area} = \dfrac{1}{2} \times d_1 \times d_2 $
where $d_1$ and $d_2$ are the diagonals of the rhombus.

Step 1: Use the given area

Given: Area = 96
So,
$96 = \dfrac{1}{2} \times d_1 \times d_2$
$\Rightarrow d_1 \times d_2 = 96 \times 2 = 192$

Step 2: Use Pythagoras Theorem

Each half-diagonal forms a right-angled triangle with the side of the rhombus.
Given side of rhombus = 10 units.

Using:
$ \left( \dfrac{d_1}{2} \right)^2 + \left( \dfrac{d_2}{2} \right)^2 = 10^2 $
$ \Rightarrow \dfrac{d_1^2}{4} + \dfrac{d_2^2}{4} = 100 $
$ \Rightarrow \dfrac{d_1^2 + d_2^2}{4} = 100 $
$ \Rightarrow d_1^2 + d_2^2 = 400 $

Step 3: Use identity for square of sum

Using identity: $ (a + b)^2 = a^2 + b^2 + 2ab $
$ (d_1 + d_2)^2 = d_1^2 + d_2^2 + 2d_1d_2 $

Substituting values:
$ (d_1 + d_2)^2 = 400 + 2 \times 192 $
$ = 400 + 384 = 784 $
$ \Rightarrow d_1 + d_2 = \sqrt{784} = 28 $

Step 4: Cost of laying electric wires

Cost per meter = ₹125 
Total length of wires = $d_1 + d_2 = 28$ meters 
Total cost = $28 \times 125 = ₹3500$

Final Answer:

The total cost of laying electric wires along the diagonals is ₹3500.