Dymond is painting a rectangular banner for a football game. She has enough paint to cover 120 ft^2. She wants the length of the banner to be 7 ft longer than the width. What dimensions should Dymond use for the banner?
w(w+7) = 120
8(15) = 120
Dymond is painting a rectangular banner for a football
game. She has enough paint to cover 120 f t 2 . She wants the
length of the banner to be 7 ft longer than the width. What
dimensions should Dymond use for the banner ?
Let's assume the width of the banner is "x" feet. Since the length is 7 feet longer than the width, the length would be "x + 7" feet.
To find the area of the rectangular banner, we multiply the length and width:
Area = Length x Width = (x + 7) ft * x ft = x^2 + 7x ft^2
Since Dymond has enough paint to cover 120 ft^2, we can set up the equation:
x^2 + 7x = 120
Now, let's solve this equation to find the dimensions of the banner.
To find the dimensions of the rectangular banner, let's assign a variable to the width. Let's call it "x" (in feet).
According to the given information, the length of the banner is 7 ft longer than the width. So, the length can be represented as: x + 7.
The area of a rectangle is calculated by multiplying the length and width. In this case, the area is given as 120 ft². So we have the equation:
Length × Width = 120
Substituting the expressions for length and width:
(x + 7) × x = 120
To solve the equation, we distribute the x:
x^2 + 7x = 120
Rearranging the equation and setting it equal to zero:
x^2 + 7x - 120 = 0
Now we can factor the quadratic equation to find its roots. The factors will help us find the width (x) of the banner:
(x + 15)(x - 8) = 0
Setting each factor to zero and solving for x:
x + 15 = 0 --> x = -15 (Discarding as we need a positive width)
x - 8 = 0 --> x = 8
We find that the width of the banner is 8 ft.
Now to find the length, we substitute this value back into the expression for length:
Length = x + 7 = 8 + 7 = 15 ft.
Therefore, the dimensions of the banner should be 8 feet wide and 15 feet long.