A rectangular auditorium seats 900 people. The number of seats in each row exceeds the number of rows by 11. Find the number of seats in each row.
Total # of seats = 900
# of rows = R
# of seats = S
So according to the question, if we have R rows then we have R+11 seats
Total # of seats = # of rows * # of seats/row
900 = R * (R+11)
900 = R^2 + 11R
0 = R^2 + 11R -900
0 = (R+36)(R-25)
Therefore, you must have 25 rows with 36 seats in each row. We can't use -36 rows because it does not give us a realistic answer.
Essentially we've created a quadratic equation and we're solving it using factoring. If you don't understand how I solved this question, I can expand on my thinking. Let me know if it makes sense.
Let's start by assigning variables to the unknown values. Let's call the number of seats in each row "x" and the number of rows "y".
According to the problem, the number of seats in each row exceeds the number of rows by 11. So, we can set up the equation:
x = y + 11
We also know that the rectangular auditorium seats 900 people. This means that the total number of seats in the auditorium can be found by multiplying the number of rows by the number of seats in each row:
Total number of seats = x * y
We can now substitute the value of "x" from the first equation into the second equation:
Total number of seats = (y + 11) * y
Since the total number of seats is given as 900, we have:
900 = (y + 11) * y
Now we can solve this quadratic equation for y. By rearranging the equation, we get:
y^2 + 11y - 900 = 0
This quadratic equation can be factored into the following form:
(y + 36)(y - 25) = 0
Setting each factor equal to zero, we have:
y + 36 = 0 or y - 25 = 0
Solving these equations, we find:
y = -36 or y = 25
Since the number of rows cannot be negative, we can disregard the first solution. Therefore, the number of rows is 25.
We can now substitute this value of "y" back into the first equation to find the number of seats in each row:
x = y + 11
x = 25 + 11
x = 36
So, the number of seats in each row is 36.
To find the number of seats in each row, let's assume the number of rows is x.
According to the given information, the number of seats in each row exceeds the number of rows by 11. So, the number of seats in each row is x + 11.
Since there are 900 people seated in the auditorium, the total number of seats can be calculated by multiplying the number of rows (x) by the number of seats in each row (x + 11).
So, the equation is:
x * (x + 11) = 900
This is a quadratic equation, which can be solved by factoring, completing the square, or using the quadratic formula. Let's solve it by factoring.
Rearranging the equation:
x^2 + 11x - 900 = 0
Now, let's factorize this quadratic equation:
(x + 36)(x - 25) = 0
Setting each factor equal to zero:
x + 36 = 0 or x - 25 = 0
Solving each equation:
x = -36 or x = 25
Since the number of rows cannot be negative, we discard the solution x = -36.
Therefore, the number of rows (x) is 25.
Now, we can find the number of seats in each row:
Number of seats in each row = x + 11 = 25 + 11 = 36
So, there are 36 seats in each row.