A rectangular auditorium seats 1716
people. The number of seats in each row exceeds the number of rows by 19
. Find the number of seats in each row.
Let's represent the number of rows as "x" and the number of seats in each row as "y".
According to the problem, the total number of seats in the auditorium is 1716, so we have the equation:
xy = 1716 (Equation 1)
The problem also states that the number of seats in each row exceeds the number of rows by 19, so we have:
y = x + 19 (Equation 2)
Now we will substitute Equation 2 into Equation 1:
(x + 19)x = 1716
Expanding the equation:
x^2 + 19x = 1716
Rearranging the equation to standard quadratic form:
x^2 + 19x - 1716 = 0
Solving this quadratic equation, we can factorize it as:
(x - 36)(x + 55) = 0
So, either x - 36 = 0 or x + 55 = 0.
If x - 36 = 0, then x = 36.
If x + 55 = 0, then x = -55. However, we cannot have a negative number of rows.
Therefore, the number of rows is 36.
Substituting this into Equation 2:
y = 36 + 19
y = 55
So, the number of seats in each row is 55.
Therefore, there are 36 rows and each row has 55 seats.