Question

An auditorium has a floor area of $20,000$ square feet. The length of the auditorium is twice its width. What are the dimensions of the room?

Answer 1

Hint: Here, in the given question, we are given the area of an auditorium and it is also given that the length of the auditorium is twice its width. As its dimensions are given in the form of length and width, we can conclude that the auditorium is in the shape of a rectangle. We know that the area of a rectangle is length × width. Here, in the given question, we are given length in terms of width. So, let $b$ be the width of the auditorium and as, length is twice its width, therefore, the length of the auditorium will be $2b$. Now, we can easily find the dimensions of the auditorium using the formula of the area of the rectangle.
Formula used:
Area of rectangle = length × width.

Complete step-by-step answer:
Let’s say that dimensions of the auditorium are:
Width = $b$
Length = $l$
From the given information, we know that
The length is twice the width: $l = 2b$

The area is $20,000$ square feet: $l \times b = 20,000$
On substituting the value of length in terms of width, we get
$ \Rightarrow 2b \times b = 20,000$
On multiplication, we get
$ \Rightarrow 2{b^2} = 20,000$
$ \Rightarrow {b^2} = \dfrac{{20,000}}{2}$
On division, we get
$ \Rightarrow {b^2} = 10,000$
On squaring both sides, we get
$ \Rightarrow b = 100feet$
Now, we will find the value of length
As $l = 2b$
$ \Rightarrow l = 2 \times 100feet$
On multiplication, we get
$ \Rightarrow l = 200feet$
Therefore, the dimensions of the auditorium are: length = $200feet$ and width = $100feet$.
So, the correct answer is “length = $200feet$ and width = $100feet$”.

Note: The key concept to solve this type of question is to learn the formulas of different shapes. While solving the squaring root, write length only in positive value because length can’t be negative. Students should be careful about the unit. Take care of the calculations so as to be sure of our final answer.