Question

A triangular park abc has sides 120m, 80 m, and 50 m. a gardener wants to put a fence all around it and also plant grass inside. how much area does he need to plant? find the cost of fencing it with barbed wire at the rate of rs 20 per metre leaving a space 3 m wide for a gate on one side

Answer 1

$\bf \large \it Hey \: User!!!$

given the sides of the triangular park are 120m, 80m and 50m.

perimeter of the triangular park = 120m + 80m + 50m
= 250m

therefore it's semi-perimeter = 250/2
= 125m

$\tt \small \: area \: of \: the \: triangular \: park \: by \\ \tt \small herons \: formula = \scriptsize {\sqrt{s(s - a)(s - b)(s - c)} } \\ \tt \footnotesize = \sqrt{125(125 - 120)(125 - 80)(125 - 50)} \\ \tt \footnotesize = \sqrt{125 \times 5 \times 45 \times 75} \\ \tt \footnotesize = \sqrt{2109375} \\ \tt \small = 1452.36 {m}^{2} \:$

hence, the gardener have to plant grass in 1452.36m²

now we have to find the cost of fencing the field with a barbed wire at the rate of rs 20 per m leaving a space of 3m wide for a gate.

therefore the gardener have to fence = 250 - 3
= 247m

so total cost of fencing at the rate of rs 20 per meter = 247 × 20
= rs 4940

$\Large \rm \it \: Cheers!!!$

Answer 2

Given:

The sides of the triangular park are 120m, 80m, and 50m

Rate of fencing= Rs.20/m

To find:

The area of the triangular park and the cost of fencing the park after leaving space for the gate

Solution:

The area of the triangular park is 375√15m².

The cost of fencing it with barbed wire is Rs.4,940.

We can find the solution by following the process given below-

We know that the area of the park can be calculated by taking all three sides of the park.

Let the sides of the park be a, b, and c.

So, a=120m, b=80m, and c=50m

Now, we will use Heron's formula to find out the area of the park.

$Area \: of \: the \: triangular \: park = \sqrt{s(s - a)(s - b)(s - c)}$

and s is the semi-perimeter of the triangular park and a, b, c are its sides.

Semi-perimeter= Sum of all sides/2

=(120+80+50)/2=250/2=125m

On putting the values, we get

$Area = \sqrt{125(125 - 120)(125 - 80)(125 - 50)}$

$= \sqrt{125(5)(45)(75)}$

$= \sqrt{625 \times 5 \times 9 \times 25 \times 3}$

=25×3×5√15=125×3√15

=375√15m²

The area where the gardener can plant grass is 375√15m².

Now, the fencing will be done along the boundary of the park i.e., the perimeter minus the space for the gate.

The perimeter of the park= Sum of all sides

=120+80+50=250m

The space for the gate is 3m wide.

So, the fencing will be done for (250-3)m=247m

The cost of fencing the park= Rs.20/m

So, the cost for fencing 247m= 20×247

=Rs.4,940

Therefore, the area of the triangular park is 375√15m² and the cost of fencing it with barbed wire is Rs.4,940.