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 1161-seat theater, we need to set up an equation based on the given information.
Let's assume the number of rows is "x". According to the problem, the number in each row is 16 fewer than the number of rows, which means the number in each row is (x - 16).
Since the total number of seats can be calculated by multiplying the number of rows by the number in each row, we can set up the equation: x * (x - 16) = 1161
Now, we can solve this equation to find the value of x, which represents the number of rows.
Expanding the equation, we get: x^2 - 16x = 1161
Rearranging the equation, we obtain: x^2 - 16x - 1161 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. However, factoring might not be feasible in this case since the coefficients are large. So, let's use the quadratic formula to find the value of x.
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation x^2 - 16x - 1161 = 0,
The coefficients are:
a = 1
b = -16
c = -1161
Substituting these values into the quadratic formula, we have:
x = (-(-16) ± √((-16)^2 - 4(1)(-1161))) / (2(1))
Simplifying the equation, we get:
x = (16 ± √(256 + 4644)) / 2
x = (16 ± √4900) / 2
x = (16 ± 70) / 2
Now, we will consider both the positive and negative values to find the possible number of rows.
For the positive value:
x = (16 + 70) / 2
x = 86 / 2
x = 43
For the negative value:
x = (16 - 70) / 2
x = -54 / 2
x = -27
Since the number of rows cannot be negative, we discard the negative solution.
Therefore, the number of seats in each row of a 1161-seat theater is 43.