The seats in a theater are arranged in parallel rows that form a rectangular region. The number of seats in each row of the theater is 16 fewer than the number of rows. How many seats are in each row of a 1161-seat theater?
I have no idea how to solve this problem. Please help if you can! Thank you! :-)
n rows
(n-16) seats/row
n * (n-16) = 1161
n^2 - 16 n - 1161 = 0
well, I want factors of 1161 that differ by 16. I know sqrt 1161 = 34.07... so I geuss that around 27 would be a good guess and 27+16 = 43 and sure enough 43*27 = 1161
so
(n-43)(n+27) = 0
n = 43
and the number of seats per row = n-16 = 27
Let R = the number of rows.
Let Sb = the number of seats in each row
The seats in the theater are arranged in parallel rows that form a rectangular region.
Then, from "The number of seats in each row of the theater is 16 fewer than the number of rows" or Sb = R - 16.
The total number of seats is given by {Sr(R) = 1161
Therefore, Sr = R - 16 = 1161/R.
I leave the hard part for you.
To solve this problem, let's break it down step by step:
1. Let's represent the number of rows in the theater as "x".
2. The problem states that the number of seats in each row is 16 fewer than the number of rows. So, the number of seats in each row can be represented as "x - 16".
3. The total number of seats in the theater can be calculated by multiplying the number of rows by the number of seats in each row. So, the equation becomes: x * (x - 16) = 1161.
4. Now we have a quadratic equation. We can solve it by setting it equal to zero and factoring or by using the quadratic formula.
5. However, factoring may become a bit tedious in this case, so we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a), where a = 1, b = -16, and c = -1161.
6. Plugging in the values, we have x = (-(-16) ± √((-16)^2 - 4(1)(-1161))) / (2(1)).
7. Simplifying further, we have x = (16 ± √(256 + 4644)) / 2.
8. Continuing the calculation, we have x = (16 ± √(4900)) / 2.
9. The square root of 4900 is 70, so we have x = (16 ± 70) / 2.
10. This gives us two possible values for x: x1 = (16 + 70) / 2 = 86 / 2 = 43, and x2 = (16 - 70) / 2 = -54 / 2 = -27 (which is not a valid number of rows in this context).
11. Since the number of rows cannot be negative, we discard x2.
12. Therefore, the theater has 43 rows.
13. Now, we can find the number of seats in each row by subtracting 16 from the number of rows: 43 - 16 = 27.
14. Thus, there are 27 seats in each row of a 1161-seat theater.
In conclusion, to find the number of seats in each row of a 1161-seat theater, we followed the steps above and determined that there are 27 seats in each row.