The first row of a school concert hall has 25 seats and each row after the first has one more seat than the row before it there are 32 rows of seats write the rule or equation for the number of seats in the nth row 35 students from the class want to sit in the same row in which row can they sit together? What is the seating capacity of the concert hall? Suppose each seat in rows 1 through 11 of the concert hall costs 100 each seat in rows 12 through 22 costs 75 and each seat in rows 23 through 32 costs 50 how much money does the concert hall take in for a sold out event?
To find the rule or equation for the number of seats in the nth row, we can use the information given. The first row has 25 seats. Each subsequent row has one more seat than the previous row.
Therefore, we can say that the number of seats in the nth row can be calculated using the equation: Seats in nth row = 25 + (n - 1).
To determine in which row the 35 students can sit together, we need to find the row that has at least 35 seats available. We can substitute the number of seats required into the equation and solve for n:
Seats in nth row = 25 + (n - 1)
35 ≤ 25 + (n - 1)
35 - 25 ≤ n - 1
10 + 1 ≤ n
11 ≤ n
Therefore, the students can sit together in row 11 or any row beyond that.
To calculate the seating capacity of the concert hall, we need to find the total number of seats in all the rows. Since the number of seats in each row follows a pattern, we can use the formula for the sum of an arithmetic series to calculate the total seating capacity.
The sum of an arithmetic series can be calculated using the formula: S = (n/2) * (a + l), where S is the sum of the series, n is the number of terms, a is the first term, and l is the last term.
In this case, we have 32 rows, and the first row has 25 seats. The last row will have 25 + (32-1) seats. Using these values, we can calculate the seating capacity:
S = (32/2) * [25 + (25 + 31)]
S = 16 * (50 + 31)
S = 16 * 81
S = 1296
Therefore, the seating capacity of the concert hall is 1296 seats.
Now let's calculate the total revenue for a sold-out event by considering the different prices for each row range.
For rows 1 through 11 with 100$ tickets, there are 11 rows with 25 seats each row:
Revenue from rows 1-11 = 11 * 25 * 100
For rows 12 through 22 with 75$ tickets, there are 11 rows with 26 seats each row:
Revenue from rows 12-22 = 11 * 26 * 75
For rows 23 through 32 with 50$ tickets, there are 10 rows with 27 seats each row:
Revenue from rows 23-32 = 10 * 27 * 50
Adding up the revenue from each row range, we can calculate the total revenue:
Total Revenue = Revenue from rows 1-11 + Revenue from rows 12-22 + Revenue from rows 23-32
Total Revenue = (11 * 25 * 100) + (11 * 26 * 75) + (10 * 27 * 50)
Calculating this expression will give us the total revenue earned by the concert hall for a sold-out event.
To find the rule or equation for the number of seats in the nth row, we need to determine the pattern in the number of seats as the rows progress.
We are given that the first row has 25 seats, and each subsequent row has one more seat than the row before it. Therefore, we can observe that the number of seats in each row forms an arithmetic sequence with a common difference of 1.
So, the rule or equation for the number of seats in the nth row can be written as:
Number of seats = 25 + (n - 1)
To calculate the row in which 35 students can sit together, we need to solve the equation for the number of seats in the nth row:
25 + (n - 1) = 35
Simplifying the equation:
n - 1 = 10
n = 11
Therefore, the 35 students can sit together in the 11th row.
To determine the seating capacity of the concert hall, we need to add up the number of seats in each row. Since there are 32 rows, we can use the rule or equation to find the number of seats in the 32nd row:
Number of seats = 25 + (32 - 1) = 25 + 31 = 56
Now, we can calculate the seating capacity by summing the number of seats in each row from the 1st to the 32nd:
Seating capacity = 25 + 26 + 27 + ... + 56
The sum of an arithmetic sequence can be calculated using the formula:
Sum = (n/2) * (first term + last term)
Here, the first term is 25, the last term is 56, and the number of terms (n) is 32. Plugging these values into the formula, we get:
Seating capacity = (32/2) * (25 + 56)
Seating capacity = 16 * 81
Seating capacity = 1296
Therefore, the seating capacity of the concert hall is 1296.
To calculate the revenue from a sold-out event, we need to multiply the number of sold seats in each price category by their corresponding prices and sum them up.
For rows 1 through 11, with 100 dollars per seat, the number of seats can be calculated using the rule or equation:
Number of seats = 25 + (n - 1)
Number of seats = 25 + (11 - 1) = 25 + 10 = 35
So, there are 35 seats in each of the first 11 rows. Multiply this by the ticket price to get the revenue for this section:
Revenue for rows 1-11 = 35 seats * 100 dollars/seat * 11 rows = 38,500 dollars
For rows 12 through 22, with seats priced at 75 dollars each, we have:
Number of seats = 25 + (n - 1)
Number of seats = 25 + (22 - 1) = 25 + 21 = 46
So, there are 46 seats in each of the rows 12 through 22. Multiply this by the ticket price:
Revenue for rows 12-22 = 46 seats * 75 dollars/seat * 11 rows = 37,950 dollars
And for rows 23 through 32, with seats priced at 50 dollars each, we have:
Number of seats = 25 + (n - 1)
Number of seats = 25 + (32 - 1) = 25 + 31 = 56
So, there are 56 seats in each of the rows 23 through 32. Multiply this by the ticket price:
Revenue for rows 23-32 = 56 seats * 50 dollars/seat * 10 rows = 28,000 dollars
Finally, to calculate the total revenue, we add up the revenues from each section:
Total revenue = Revenue for rows 1-11 + Revenue for rows 12-22 + Revenue for rows 23-32
Total revenue = 38,500 dollars + 37,950 dollars + 28,000 dollars
Total revenue = 104,450 dollars
Therefore, the concert hall takes in $104,450 for a sold-out event.
you have an AP with
a = 25
d = 1
The nth row has a + (n-1)d seats
The $1.00 seats are rows 1-11, so
S11 = 1.00 * 11/2 (2*25+10*1)
Row 12 has 25+11 = 36 seats, so the 11 rows of $0.75 seats cost
0.75 * 11/2 (2*36+10*1)
and so on