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प्रश्न
| A school auditorium has to be constructed to accommodate at least 1500 people. The chairs are to be placed in a concentric circular arrangement in such a way that each succeeding circular row has 10 seats more than the previous one. |
Based on the above information, answer the following questions.
- If the first circular row has 30 seats then how many seats will there be in the 10th row?
- For 1500 seats, how many rows must the auditorium have?
- If 1500 seats are to be arranged in the auditorium, how many seats are still left to be put after the 10th row?
- If there were 17 rows in the auditorium, how many seats would be in the middle row?
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उत्तर
The seating arrangement follows an Arithmetic Progression (AP):
First term (seats in first row), a = 30
Common difference, d = 10
i. Seats in the 10th row:
Formula: \[T_n = a + (n - 1)d\]
Substitution: \[T_{10} = 30 + (10 - 1) \times 10\]
\[T_{10} = 30 + 90 = 120\]
There are 120 seats in the 10th row.
ii. Number of rows required for 1500 seats:
Given: Total sum \[S_n = 1500\]
Formula: \[S_n = \frac{n}{2}[2a + (n - 1)d]\]
Calculation: \[1500 = \frac{n}{2}[2(30) + (n - 1)10]\]
\[3000 = n[60 + 10n - 10]\]
\[3000 = n(10n + 50)\]
\[10n^2 + 50n - 3000 = 0\]
Dividing the entire equation by 10: \[n^2 + 5n - 300 = 0\]
Factoring: \[n^2 + 20n - 15n - 300 = 0\]
\[n(n + 20) - 15(n + 20) = 0\]
\[(n + 20)(n - 15) = 0\]
\[n = -20 \quad \text{or} \quad n = 15\]
Since rows cannot be negative, n = 15.
The auditorium must have 15 rows.
iii. Seats still left to be put after the 10th row:
Total seats accommodated up to 10th row:
\[S_{10} = \frac{10}{2}[2(30) + (10 - 1)10]\]
\[S_{10} = 5 \times [60 + 90]\]
\[= 5 \times 150\]
\[= 750\]
Seats left to be put: \[\text{Remaining seats} = 1500 - S_{10}\]
\[= 1500 - 750\]
\[= 750\]
750 seats are still left to be put.
iv. Number of seats in the middle row if there are 17 rows:
Middle row determination: For n = 17, the middle row is the:
\[\left(\frac{17 + 1}{2}\right)\text{th row} = 9\text{th row}\]
Seats in the 9th row: \[T_9 = a + (9 - 1)d\]
\[T_9 = 30 + 8(10)\]
\[= 30 + 80\]
\[= 110\]
There are 110 seats in the middle row.
