need help with this one
the length of a rectangular banner is 4 feet longer than its width if the area is 77 squar feet find the dimensions
Let
w=width
length=w+4
Area,
77=(w+4)w
Solve for w:
w²+4w-77=0
(w+11)(w-7)=0
So
w=7 or w=-11 (reject)
=>
width = 7'
check:
7*(7+4)=77, good.
To find the dimensions of the rectangular banner, we can use the given information about the area and the relationship between the length and width.
Let's assume the width of the banner is "x" feet.
According to the problem, the length of the banner is 4 feet longer than its width. So, the length can be represented as "x + 4" feet.
We know that the area of a rectangle is calculated by multiplying its length by its width. In this case, the area is given as 77 square feet.
Therefore, we can set up the following equation:
Length * Width = Area
(x + 4) * x = 77
Expanding the equation, we get:
x^2 + 4x = 77
Rearranging the equation to make it equal to zero, we have:
x^2 + 4x - 77 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula.
Using the quadratic formula, which states that the solutions for ax^2 + bx + c = 0 are given by the formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a),
we can find the value of x by plugging in the corresponding values. In this case, a = 1, b = 4, and c = -77.
x = (-4 ± sqrt(4^2 - 4 * 1 * -77)) / (2 * 1)
Simplifying further:
x = (-4 ± sqrt(16 + 308)) / 2
x = (-4 ± sqrt(324)) / 2
x = (-4 ± 18) / 2
This results in two potential values for x:
x1 = (-4 + 18) / 2 = 14/2 = 7
x2 = (-4 - 18) / 2 = -22/2 = -11
Since the width of a banner cannot be negative, we can disregard the second solution.
Therefore, the width of the rectangular banner is 7 feet.
Now, we can find the length by adding 4 feet to the width:
Length = Width + 4 = 7 + 4 = 11
So, the dimensions of the rectangular banner are:
Width = 7 feet
Length = 11 feet
To find the dimensions of the rectangular banner, we can set up an equation based on the given information. Let's denote the width as "w" and the length as "l."
According to the problem, the length of the banner is 4 feet longer than its width. Therefore, we can write:
l = w + 4
The area of a rectangle is calculated by multiplying the length and width. In this case, the area is given as 77 square feet:
Area = length × width
77 = l × w
Now substitute the value of the length (l) from the first equation into the area equation:
77 = (w + 4) × w
To solve this quadratic equation, we can rearrange it to the standard form:
w^2 + 4w - 77 = 0
Now we can use the quadratic formula to find the possible values of w:
w = (-b ± √(b^2 - 4ac)) / (2a)
For the equation w^2 + 4w - 77 = 0, the values of a, b, and c are:
a = 1, b = 4, and c = -77.
Using the quadratic formula, we can calculate the values of w.
w = (-4 ± √(4^2 - 4(1)(-77))) / (2(1))
w = (-4 ± √(16 + 308)) / 2
w = (-4 ± √324) / 2
w = (-4 ± 18) / 2
Simplifying further, we get two possible values for w:
w1 = (-4 + 18) / 2 = 14 / 2 = 7
w2 = (-4 - 18) / 2 = -22 / 2 = -11
Since the width cannot be negative, the value w = -11 is not valid. Hence, the width of the rectangular banner is 7 feet.
Now, substitute the value of w back into the first equation to find the length:
l = w + 4 = 7 + 4 = 11
Therefore, the dimensions of the rectangular banner are width = 7 feet and length = 11 feet.