Auditorium Seating and the Lawn Walkway
An auditorium has 25 rows. The first row has 18 seats, and each row after that has 2 more seats than the row before. The manager claims the hall seats 1,200 people and plans to sell that many tickets. Attached to the hall is a rectangular lawn with perimeter 68 m and area 240 m²; the manager wants to lay a straight diagonal walkway across it and add a 1 m-wide border along the two longer sides. A supplier says the diagonal is 'about 25 m', which would determine the length of paving material. The finance team will approve the ticket plan only if the seating claim is accurate, and the gardener needs the lawn dimensions before ordering border tiles at ₹450 per square metre.
1. Calculate the total number of seats using the arithmetic progression, and decide whether the manager can safely sell 1,200 tickets.
Read the worked answer
The row counts form an AP with first term a = 18, common difference d = 2, and n = 25.
Last row: a₂₅ = 18 + (25 − 1) × 2 = 18 + 48 = 66 seats.
Sum = n/2 × (first + last) = 25/2 × (18 + 66) = 12.5 × 84 = 1,050 seats.
The hall seats 1,050, not 1,200. Selling 1,200 tickets would oversell by 150 seats, so the claim is inaccurate and the ticket plan should not be approved as stated.
2. Find the dimensions of the rectangular lawn and the exact diagonal length. Assess the supplier's 'about 25 m' estimate.
Read the worked answer
Let the sides be x and y metres. Then 2(x + y) = 68, so x + y = 34, and xy = 240.
The sides are roots of t² − 34t + 240 = 0. Discriminant = 34² − 4 × 240 = 1,156 − 960 = 196, and √196 = 14.
t = (34 ± 14)/2, giving 24 and 10. The lawn is 24 m by 10 m.
Diagonal = √(24² + 10²) = √(576 + 100) = √676 = 26 m. The supplier's 25 m is 1 m short, which is a material error when ordering paving.
3. Calculate the area of the 1 m-wide border along the two longer sides and its cost at ₹450 per square metre. State any assumption you make about corners.
Read the worked answer
The longer sides are 24 m each. A 1 m-wide strip along each longer side has area 24 × 1 = 24 m². For two such strips the area is 48 m².
Cost = 48 × 450 = ₹21,600.
This assumes the two strips are on opposite longer sides and do not overlap at corners; if the border turns corners, the corner squares would need separate treatment. Under the stated 'along the two longer sides' plan, 48 m² is the defensible area.