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?
S = # of seats per row
R = # of rows
S = R - 16
S*R = R * (R-16) = 1161
R^2 - 16R - 1161 = 0
1161 is the product of 43 and 27. The equation factors to
(R - 43)(R + 27) = 0
R = 43 is the only positive answer for the number of rows. There are
S = R - 16 = 27 seats per row.
To find the number of seats in each row of a 1161-seat theater, we can set up an equation using the given information.
Let's assume the number of rows in the theater is "x". According to the given information, the number in each row is 16 fewer than the number of rows. Therefore, the number of seats in each row can be represented as "x - 16".
Since the total number of seats in the theater is 1161, we can set up an equation:
Number of rows * Number in each row = Total number of seats
x * (x - 16) = 1161
Now, we can solve this equation to find the value of x:
x^2 - 16x = 1161
x^2 - 16x - 1161 = 0
This is a quadratic equation that can be solved by factoring, using the quadratic formula, or by completing the square. Let's solve it by factoring:
(x - 51)(x + 23) = 0
Setting each factor equal to zero, we get:
x - 51 = 0 or x + 23 = 0
Solving for x, we find two possible solutions:
x = 51 or x = -23
Since the number of rows cannot be negative, we can ignore x = -23. Therefore, the number of rows in the theater is 51.
To find the number of seats in each row, we can substitute this value back into the equation:
Number of seats in each row = x - 16 = 51 - 16 = 35
So, there are 35 seats in each row of the 1161-seat theater.