The seats in a theater are arranged in parallel rows that form a rectangular region. The number in each row is 16 fewer than the number of rows. How many seats are in each row of 1161 -seat theater?
To find the number of seats in each row of a theater with 1161 seats, we need to use the given information that the number of seats in each row is 16 fewer than the number of rows.
Let's assume the number of rows in the theater is "x". So, the number of seats in each row would be x - 16.
Now, we know that the total number of seats in the theater is 1161. To calculate the number of seats in each row, we need to divide the total number of seats by the number of rows:
1161 / x = x - 16
To solve this equation, let's simplify it:
1161 = x(x - 16)
Expanding this equation, we get:
1161 = x^2 - 16x
Now, let's bring all terms to one side of the equation:
x^2 - 16x - 1161 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. Factoring might not be straightforward in this case, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = -16, and c = -1161. Plugging these values into the formula, we get:
x = (-(-16) ± √((-16)^2 - 4 * 1 * -1161)) / (2 * 1)
Simplifying further:
x = (16 ± √(256 + 4644)) / 2
x = (16 ± √4900) / 2
x = (16 ± 70) / 2
Now, let's calculate the two possible values of x:
x1 = (16 + 70) / 2 = 86 / 2 = 43
x2 = (16 - 70) / 2 = -54 / 2 = -27
Since the number of rows cannot be negative, we discard the negative value of x.
Therefore, the number of rows in the theater is 43.
To find the number of seats in each row, we substitute this value back into the equation x - 16:
Number of seats in each row = 43 - 16 = 27
So, there are 27 seats in each row of a 1161-seat theater.