A hockey arena has 10920 seats. The first row of seats around the rink has 220 seats. The number of seats in each subsequent row increases by 16. How many rows of seats does the arena have?
n/2 (2*220 + (n-1)*16) = 10920
solve for n
To find the number of rows of seats in the arena, we can use the information given.
Let's say the number of rows of seats is represented by "n".
We know that the first row has 220 seats.
The number of seats in each subsequent row increases by 16.
To find the total number of seats, we can use the formula for the sum of an arithmetic series:
S = (n/2)(2a + (n-1)d)
Where:
S = total number of seats (10920)
a = first term (220)
n = number of terms (unknown)
d = common difference (16)
Substituting the given values into the formula, we have:
10920 = (n/2)(2(220) + (n-1)(16))
Now let's simplify and solve for n:
10920 = (n/2)(440 + 16n - 16)
10920 = (n/2)(424 + 16n)
10920 = (n/2)(16n + 424)
10920 = 8n^2 + 212n
Rearranging the equation, we have:
8n^2 + 212n - 10920 = 0
Now we can solve this quadratic equation. Using factoring, completing the square, or the quadratic formula, we find that n is approximately equal to 12.
Therefore, the hockey arena has 12 rows of seats.
To find the number of rows of seats in the hockey arena, we can use the given information about the first row and the increase in seats for each subsequent row.
First, let's determine the number of seats in the rows beyond the first row. We are told that each subsequent row has 16 more seats than the previous row.
Since the first row has 220 seats, the second row will have 220 + 16 = 236 seats.
The third row will have 236 + 16 = 252 seats.
The fourth row will have 252 + 16 = 268 seats.
And so on.
Now, we need to figure out when the number of seats in a row exceeds the total number of seats in the arena, which is 10920.
Let's set up an equation:
220 + 236 + 252 + 268 + ... + (220 + 16*(n-1)) > 10920.
Simplifying the equation, we get:
220*n + 16*(1+2+3+...+n-1) > 10920.
Since the sum of the first n natural numbers can be expressed as n*(n-1)/2, the equation becomes:
220*n + 16*(n*(n-1)/2) > 10920.
Simplifying further:
220n + 8n(n-1) > 10920.
220n + 8n^2 - 8n > 10920.
Rearranging the equation:
8n^2 + 212n - 10920 > 0.
To solve this quadratic inequality, we can use the quadratic formula, which states that for an equation ax^2 + bx + c = 0, the roots are given by:
x = (-b ± sqrt(b^2 - 4ac)) / 2a.
For our quadratic inequality, a = 8, b = 212, and c = -10920. Plugging these values into the quadratic formula, we get:
n = (-212 ± sqrt(212^2 - 4*8*(-10920))) / (2*8).
n = (-212 ± sqrt(44944 + 350400)) / 16.
n = (-212 ± sqrt(395344)) / 16.
n = (-212 ± 628) / 16.
Considering the positive root, n = (-212 + 628) / 16.
n = 416 / 16.
n = 26.
Therefore, the hockey arena has 26 rows of seats.