Setting up word problem for finding length and width

Question

Word Problem: The length of a rectangular sign is $3$ feet longer than the width. If the sign has space for $54$ square feet of advertising, find its length and width.

I have not idea where to start. What is the formula to solve this?

Answer 1

The sides of the rectangle are $x$ (width) and $y$ (length). Now, write down:

1. The equation that tells you that the sign is $3$ feet longer than it is wide.
2. The equation that tells you that the area of the sign is $54$ square feet.

Put these equations either in your question via editing or in the comments under my answer, I will then help you further.

Answer 2

Let $w$ be the width of the sign. It is given that the length is $w+3$, and therefore a variable for length is not necessary (though this is essentially just skipping a step of solving the system of equations as given above). Area=width * length, so $54=w(w+3)$. Solving for $w$ is now trivial, you already have a formula for length in terms of $w$.

Answer 3

Sol. 1. Brute force it. Basically, try product of all numbers between 1 and 54. And you will find that the answer is 6 x 9 = 54.

But that trivial so lets see if we can improve.

Sol. 2. Let breath of sign board = x. This means length of sign board should be = (x+3)

So our solution will be the value of x such that x * (x + 3) = 54 Now either you can solve the quadratic equation x^2 + 3x = 54. If you don't know how to solve quadratic equation, that try each value of x till you get x * (x + 3 ) = 54. ( an easier way to brute force :) ). And x = 6 will satisfy the equation.

Hence length = 9 feet and breath = 6 feet.