A community park wants to place a rectangular billboard to inform guests of their new attractions. Suppose the length of the billboard to be placed is 5m longer than its width and the area is 500m2. What will be the length and the width of the billboard?
Let w be the width of the billboard
L = w + 5
w * (w + 5) = 500
w² + 5 w - 500 = 0
factoring ... (w + 25) (w - 20) = 0
w = 20 ... L = 25
To solve this question, we can use a system of equations.
First, we'll define the width of the billboard as 'w'. Since the length of the billboard is 5 meters longer than its width, we can say that the length is 'w + 5'.
Next, we know that the area of a rectangle is equal to the length multiplied by the width. In this case, the area is given as 500 square meters. So we can set up the equation:
w * (w + 5) = 500
To solve this equation, we can expand it:
w^2 + 5w = 500
Rearrange the equation to make it quadratic:
w^2 + 5w - 500 = 0
Now, we can either solve this equation by factoring, completing the square, or using the quadratic formula. Since it doesn't seem easily factorable and completing the square may be a bit more involved, let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / 2a
In our equation, a = 1, b = 5, and c = -500. Plugging these values into the quadratic formula, we get:
w = (-5 ± √(5^2 - 4 * 1 * -500)) / (2 * 1)
Simplifying:
w = (-5 ± √(25 + 2000)) / 2
w = (-5 ± √(2025)) / 2
w = (-5 ± 45) / 2
So we have two possible solutions for the width, w1 = (-5 + 45) / 2 = 20/2 = 10 and w2 = (-5 - 45) / 2 = -50/2 = -25. Since the width cannot be negative in this context, we discard the negative solution.
Therefore, the width of the billboard is 10 meters (w = 10).
Since the length is defined as the width plus 5, the length is 10 + 5 = 15 meters.
So the length of the billboard is 15 meters and the width is 10 meters.