Cost of electric wires along rhombus diagonals

CAT 2021 Slot 3 · QA · Easy · Geometry

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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

3500

Explanation

Perimeter of rhombus = 40 m   ⟹  \implies Side length s=10s = 10 m.

Let the diagonals be d1d_1 and d2d_2. Area=12d1d2=96  ⟹  d1d2=192\text{Area} = \frac{1}{2} d_1 d_2 = 96 \implies d_1 d_2 = 192

Also, (d12)2+(d22)2=s2=100  ⟹  d12+d22=400\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = s^2 = 100 \implies d_1^2 + d_2^2 = 400.

(d1+d2)2=d12+d22+2d1d2=400+384=784  ⟹  d1+d2=28(d_1 + d_2)^2 = d_1^2 + d_2^2 + 2d_1 d_2 = 400 + 384 = 784 \implies d_1 + d_2 = 28

Total length of electric wires along two diagonals = d1+d2=28d_1 + d_2 = 28 m. Total cost = 28×125=₹350028 \times 125 = ₹ 3500.

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