Ellen is making rectangular bookmarks. Each bookmark has an area of 16 square inches, and the length is 6 inches more than the width. Find the length and width of her bookmarks.
Let x = width
Let x + 6 = length
Area of rectangle is just width x length. Thus,
A = w * l
16 = x(x+6)
16 = x^2 + 6x
x^2 + 6x - 16 = 0
Factor:
(x + 8)(x - 2) = 0
x = -8 (extraneous root since dimensions cannot be negative)
x = 2 inches (width)
x+6 = 8 inches (length)
hope this helps~ `u`
Let's denote the width of the bookmark as "x" inches.
According to the given information, the length of the bookmark is 6 inches more than the width. Therefore, the length of the bookmark is (x + 6) inches.
The area of the rectangle is equal to the length multiplied by the width. So, we have the equation:
Area = Length × Width
Using the given area of 16 square inches, we can write the equation as:
16 = (x + 6) × x
Now, let's solve the equation:
16 = x^2 + 6x
Rearranging the equation to standard form:
x^2 + 6x - 16 = 0
To solve the quadratic equation, we can factor it or use the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For the equation x^2 + 6x - 16 = 0, the coefficients are:
a = 1, b = 6, and c = -16.
Substituting these values into the quadratic formula:
x = (-(6) ± sqrt((6)^2 - 4(1)(-16))) / (2(1))
Simplifying:
x = (-6 ± sqrt(36 + 64)) / 2
x = (-6 ± sqrt(100))/2
x = (-6 ± 10) / 2
So, we have two possible solutions for the width of the bookmark:
x = (-6 + 10) / 2 = 4/2 = 2 inches
x = (-6 - 10) / 2 = -16 / 2 = -8 inches (Ignore this negative value as width cannot be negative)
Therefore, the width of the bookmark is 2 inches.
Now, let's find the length by substituting the value of x into the equation:
Length = x + 6 = 2 + 6 = 8 inches
So, the length and width of Ellen's bookmark are 8 inches and 2 inches, respectively.
To find the length and width of Ellen's rectangular bookmarks, we can use the information given regarding the area and the relationship between the length and the width.
Let's assume the width of the bookmark is x inches. According to the question, the length of the bookmark is 6 inches more than the width. So, the length would be x + 6 inches.
To find the area of a rectangle, we multiply the length by the width. Therefore, the area of the bookmark is:
Area = Length × Width
Given in the question, the area is 16 square inches. Plugging in the values, we have:
16 = (x + 6) × x
Now, let's solve this equation to find the values of x (width) and x + 6 (length).
To simplify the equation, let's expand the right side:
16 = x^2 + 6x
Rearranging the equation to form a quadratic equation:
x^2 + 6x - 16 = 0
To solve this equation, we can factor it or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation x^2 + 6x - 16 = 0, the coefficients are:
a = 1 (coefficient of x^2)
b = 6 (coefficient of x)
c = -16 (constant term)
Plugging these values into the quadratic formula, we have:
x = (-6 ± √(6^2 - 4(1)(-16))) / (2*1)
Simplifying further:
x = (-6 ± √(36 + 64)) / 2
x = (-6 ± √100) / 2
x = (-6 ± 10) / 2
Now we have two possible values for x:
1. x = (-6 + 10) / 2 = 4/2 = 2
2. x = (-6 - 10) / 2 = -16/2 = -8
Since dimensions cannot be negative, we discard the negative value for x.
Therefore, the width (x) of the bookmark is 2 inches.
Now, to find the length:
Length = Width + 6 = 2 + 6 = 8 inches
Hence, the length of the bookmark is 8 inches, and the width is 2 inches.