Triangular Ground Fencing and Stage
A school sports committee has a triangular open ground with sides 13 m, 14 m, and 15 m. They plan to fence the boundary, leaving a 3 m gate on the 14 m side. Fencing costs ₹185 per metre. Inside the ground, they want a circular stage of radius 2 m for an assembly. A member claims the stage will fit without touching any side because 'the triangle is big enough', but another wants a calculation. The committee also needs to know how much grass area remains after the stage is built, assuming the stage is a full circle and no grass grows under it. A local supplier offers a 7% discount on the fencing cost if the order is above ₹5,000. The committee must decide whether the discounted fencing cost plus stage work is within a ₹9,000 budget; stage work is quoted at ₹1,250.
1. Use Heron’s formula to find the area of the triangular ground and its inradius. Decide whether the 2 m radius stage fits without crossing any side.
Read the worked answer
Semiperimeter s=(13+14+15)/2=21 m. Area=√[21(21-13)(21-14)(21-15)] =√(21×8×7×6) =√7056=84 m².
Inradius r=Area/s=84/21=4 m. The largest circle that can fit inside the triangle has radius 4 m. Since 2 m<4 m, the stage fits without touching any side. The member’s claim is correct, but the calculation is needed to justify it.
2. Calculate the fencing length and cost after the 7% discount. Including stage work, decide whether the total is within the ₹9,000 budget.
Read the worked answer
Perimeter=13+14+15=42 m. Gate=3 m, so fencing length=42-3=39 m. Cost before discount=39×185=₹7,215. The order is above ₹5,000, so the discount applies. Discount=7% of 7215=0.07×7215=₹505.05. Discounted fencing cost=7215-505.05=₹6,709.95.
Total with stage work=6709.95+1250=₹7,959.95. This is less than ₹9,000, so the plan is within budget by about ₹1,040.
3. Find the area of the circular stage and the grass area remaining. Use π≈22/7 and give the answer to the nearest square metre.
Read the worked answer
Stage area=πr²=π(2²)=4π. Using π≈22/7: Stage area≈4×22/7=88/7≈12.57 m².
Remaining grass area=triangle area-stage area ≈84-12.57=71.43 m². To the nearest square metre, the remaining grass area is about 71 m².