A rectangular auditorium seats 1862 people. The number of seats in each row exceeds the number of rows by 11. Find the number of seats in each row.
Let the number of rows be $r$. Since the number of seats in each row exceeds the number of rows by 11, there must be $r+11$ seats in each row. Thus, we have $r(r+11)=1862$. Factoring 1862, we quickly find the prime factorization 1862=$2 \cdot 19 \cdot 49$. $49=7^2$, so $r=7$. Our answer is $r+11=\boxed{18}$.
A rectangular auditorium seats 1862 people. The number of seats in each row exceeds the number of rows by 11. Find the number of seats in each row.
Let $r$ be the number of rows and $s$ be the number of seats in each row. We know that $s = r + 11$ and that $rs = 1862$. Substituting the first equation into the second equation, we have $(r+11)r = 1862$. Expanding the left side, we get $r^2 + 11r = 1862$. Rearranging this equation, we have $r^2 + 11r - 1862 = 0$. We can factor this into $(r + 34)(r - 23) = 0$. This gives us two possible values for $r$: $r = -34$ and $r = 23$. Since the number of rows cannot be negative, we have $r = 23$. Plugging this into the first equation, we have $s = 23 + 11 = 34$. Therefore, there are $\boxed{34}$ seats in each row.
Let's start by assigning variables to the information given.
Let:
x = number of rows
x + 11 = number of seats in each row
The total number of seats is the product of the number of rows and the number of seats in each row, which is given as 1862.
So, we have the equation:
x * (x + 11) = 1862
To solve this equation, we can simplify it by expanding the product:
x^2 + 11x = 1862
Next, we can move all terms to one side of the equation to have a quadratic equation set equal to zero:
x^2 + 11x - 1862 = 0
Now, we can solve this quadratic equation. We can either factor this equation or use the quadratic formula.
Using the quadratic formula, where a = 1, b = 11, and c = -1862, the quadratic formula is:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Substituting the values into the quadratic formula, we have:
x = (-11 ± sqrt(11^2 - 4*1*(-1862))) / (2*1)
Simplifying further:
x = (-11 ± sqrt(121 + 7448)) / 2
x = (-11 ± sqrt(7569)) / 2
x = (-11 ± 87) / 2
Using both solutions, we have:
x = (-11 + 87) / 2
x = 76 / 2
x = 38
and
x = (-11 - 87) / 2
x = -98 / 2
x = -49
Since the number of rows cannot be negative, we discard the negative solution.
Therefore, the number of rows is 38.
To find the number of seats in each row, we can substitute the value of x back into the equation x + 11.
Number of seats in each row = 38 + 11
Number of seats in each row = 49
So, there are 49 seats in each row.
Let's break down the problem into smaller steps to find the solution.
Step 1: Assign variables.
Let's assign variables to the unknowns in the problem.
Let's say the number of rows is 'x'.
Since the number of seats in each row exceeds the number of rows by 11, the number of seats in each row can be represented as 'x + 11'.
Step 2: Set up the equation.
The formula for finding the total number of seats in an auditorium is given by the product of the number of rows and the number of seats in each row. In this case, it will be 'x * (x + 11)'.
Step 3: Solve the equation.
According to the problem, the total number of seats in the auditorium is 1862. We can now set up the equation:
x * (x + 11) = 1862
To solve this quadratic equation, we can simplify it further:
x^2 + 11x = 1862
Rearranging the equation to the standard quadratic form:
x^2 + 11x - 1862 = 0
Step 4: Solve the quadratic equation.
We can solve the quadratic equation by factoring, completing the square, or using the quadratic formula.
In this case, factoring might not be straightforward, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation: x^2 + 11x - 1862 = 0
The coefficients are: a = 1, b = 11, and c = -1862.
Plugging these values into the quadratic formula:
x = (-11 ± √(11^2 - 4*1*(-1862))) / (2*1)
Simplifying:
x = (-11 ± √(121 + 7448)) / 2
x = (-11 ± √(7569)) / 2
x = (-11 ± 87) / 2
Solving for x, we get two values:
x1 = (-11 + 87) / 2 = 76 / 2 = 38
x2 = (-11 - 87) / 2 = -98 / 2 = -49
Since the number of rows cannot be negative, we discard x2 = -49.
Step 5: Find the number of seats in each row.
We know that the number of seats in each row is equal to 'x + 11'.
So substituting the value of x from the above step, we get:
Number of seats in each row = 38 + 11 = 49
Therefore, the number of seats in each row is 49.