Question

A rectangular auditorium seats 3300 people. The number of seats in each row exceeds the number of rows by 16. Find the number of seats in eachrow.

Answer 1

Take x as the number of rows and y as the number of seats in each row.

There are 3300 seats in the auditorium, which means that the product of x and y is 3300:

$x\cdot y=3300$

We also know that y exceeds x by 16, this is:

$y=x+16$

Use these equations to find the number of seats and rows.

$\begin{gathered} x\cdot(x+16)=3300 \\ x^2+16x=3300 \\ x^2+16x-3300=0 \end{gathered}$

Use the quadratic equation to find x:

$\begin{gathered} x=\frac{-16\pm\sqrt[]{16^2-4(1\cdot-3300)}}{2\cdot1} \\ x=\frac{-16\pm116}{2} \\ x1=\frac{-16+116}{2}=\frac{100}{2}=50 \\ x2=\frac{-16-116}{2}=\frac{-132}{2}=-66 \end{gathered}$

As x is the number of rows, it can take negative values, so take the value of 50.

Use this values to find y:

$\begin{gathered} x\cdot y=3300 \\ y=\frac{3300}{x} \\ y=\frac{3300}{50} \\ y=66 \end{gathered}$

There are 50 seats in each row and there are 66 rows.