Question

The area of a triangular field with equal sides is equal to 3√ 5 times the area of another triangular park whose sides are 50 m, 80 m and 120 m. what will be the cost of putting the fence around the field at the rate of Rs. 105 per metre?
1. Rs. 46,830
2. Rs. 47,460
3. Rs. 47,250
4. Rs. 46,935

Answer 1

Correct Answer - Option 3 : Rs. 47,250

Given:

Area of an equilateral triangular field ΔABC = 3√5 × area of triangular park ΔPQR

Dimensions of ΔPQR = 50m, 80m and 120m

The cost of the fence is Rs. 105 per metre

Formula used:

Area of a triangle = \(√{s(s - a)(s - b)(s - c)}\)

Perimeter = a + b + c

Semi perimeter(s) = \(\frac{a \ + \ b \ +\ c}{2}\)

Area of equilateral triangle = \(\frac{√3}{4}a^2\)

Where a, b and c are sides of a triangle

Calculation:

First find the area of ΔPQR

s = \(\frac{50 \ + \ 80 \ +\ 120}{2}\)

⇒ s = 125 metre

Area of the ΔPQR

\(√{125(125 - 50)(125 - 80)(125 - 120)}\)

⇒ \(√{125 × 75 × 45 × 5}\)

⇒ \(√{25 × 5 × 25 × 3× 5 × 9 × 5}\)

⇒ 5 × 5 × 5 × 3 × √15

⇒ 375√15 m²

Area of triangle ΔABC

375√15 × 3√5 = 5625√3 m²

Side of ΔABC

\(\frac{√3}{4}a^2\) = 5625√3

⇒ a² = 22500

⇒ a = 150 m

Perimetre of ΔABC is

⇒ 3 × 150 = 450 m

Cost of fencing 450 m is

450 × 105 = Rs. 47,250

∴ The cost of fencing field is Rs. 47,250.