The first row of a concert hall has 20 seats, and each row after the first has one more seat than the row before it. There are 30 seats.
a)write a rule for the number of seats in the nth row
b) 40 students from a class want to sit n the same row.how close can they sit?
For (A) I got n+19 and for (B) I got 21
S(n) = 20 + (n-1) which is indeed n+19
n+19 = 40
so
n = 21
To find the rule for the number of seats in the nth row, we need to observe the pattern mentioned. Let's break it down step-by-step:
a) Write a rule for the number of seats in the nth row:
In the given concert hall, the first row has 20 seats. Each row after the first has one more seat than the row before it. This indicates that the number of seats in each row increases by 1 as we move to the next row.
We can represent this pattern mathematically using n, the row number, as follows:
Number of seats in the nth row = 20 + (n-1)
So, the rule for the number of seats in the nth row is n + 19.
b) How close can 40 students from a class sit in the same row?
If 40 students want to sit in the same row, we need to find the row that can accommodate the maximum number of students. Let's work it out:
Using the rule we found in part A:
Number of seats in the nth row = n + 19
Now, we need to find the row number (n) that can accommodate at least 40 students:
n + 19 >= 40
Solving this inequality:
n >= 40 - 19
n >= 21
Therefore, the minimum row number (n) required to seat 40 students is 21.
Hence, the closest the 40 students can sit is in the 21st row.
Great job on your answers for (A) and (B)!
To further explain:
(a) To find the rule for the number of seats in the nth row, we can observe that the first row has 20 seats. As we move to the second row, it has one more seat than the first row, which means it has 20 + 1 = 21 seats. For the third row, it would have 21 + 1 = 22 seats, and so on.
Therefore, the general rule for the number of seats in the nth row is n + 19.
(b) If 40 students from the class want to sit in the same row, we need to find the row with the closest available number of seats to accommodate all of them.
We can start by substituting different values for n into the formula n + 19 and see which row gives us a number closest to or greater than 40.
For n = 21, the number of seats in the 21st row would be 21 + 19 = 40 seats. This means that if the class wants to sit in the same row, they would need at least the 21st row.
So, your answer of 21 is correct, which means the closest the 40 students can sit together is in the 21st row.