The width of a rectangle is 8 less than twice its length. If the area of the rectangle is 93
cm
2
, what is the length of the diagonal?
If its length is x, then its width is 2x-8
x(2x-8) = 93
solve for x, and then use the Pythagorean Theorem to find the diagonal.
L = Length
W = Width
A = Area
The width of a rectangle is 8 less than twice its length means:
W = 2 L - 8
A = L ∙ W
93 = L ∙ ( 2 L - 8 )
93 = 2 L² - 8 L
Subtract 93 to both sides
0 = 2 L² - 8 L - 93
Now solve equation:
2 L² - 8 L - 93 = 0
The solutions are:
L = 2 - √ ( 101 / 2 ) = - 5.1063 cm
and
L = 2 + √ ( 101 / 2 ) = 9.1063 cm
Length cannot be negative, so:
L = 2 + √ ( 101 / 2 )
W = 2 L - 8
W = 2 [ 2 + √ ( 101 / 2 ) ] - 8
W = 4 + 2 √ ( 101 / 2 ) - 8
W = 2 √ ( 101 / 2 ) - 4
W = √4 ∙ √ ( 101 / 2 ) - 4
W = (√ 4 ∙ 101 / 2 ) - 4
W = (√ 404 / 2 ) - 4
W = √202 - 4
W = 14.21267 - 4 = 10.21267 cm
Length of the diagonal:
d = √ ( L² + W² )
d = √ ( 9.1063² + 10.21267² )
d = 13.633 cm
To find the length of the diagonal of a rectangle, we need to use the Pythagorean theorem.
Let's call the length of the rectangle L and the width W.
Given that the width of the rectangle is 8 less than twice its length, we can write the equation:
W = 2L - 8
The formula for the area of a rectangle is:
Area = Length * Width
We know that the area of the rectangle is 93 cm^2, thus:
93 = L * W
Substituting the value of W from the previous equation, we have:
93 = L * (2L - 8)
Expanding the equation, we get:
93 = 2L^2 - 8L
Rearranging the equation, we have:
2L^2 - 8L - 93 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula.
In this case, the equation doesn't factor easily, so we can use the quadratic formula:
L = (-b ± sqrt(b^2 - 4ac)) / 2a
For our equation, a = 2, b = -8, and c = -93. Substituting these values, we get:
L = (-(-8) ± sqrt((-8)^2 - 4(2)(-93))) / 2(2)
Simplifying inside the square root, we have:
L = (8 ± sqrt(64 + 744)) / 4
L = (8 ± sqrt(808)) / 4
L = (8 ± 28.42) / 4
Now, let's consider both possibilities:
1) L = (8 + 28.42) / 4 = 36.42 / 4 = 9.11 cm
2) L = (8 - 28.42) / 4 = -20.42 / 4 = -5.11 cm
Since the length cannot be negative, we disregard the second value.
So, the length of the rectangle is 9.11 cm.
To find the diagonal, we use the Pythagorean theorem:
Diagonal^2 = Length^2 + Width^2
Let's substitute the values we have:
Diagonal^2 = (9.11)^2 + (2(9.11) - 8)^2
Diagonal^2 = 83.0321 + (10.22)^2
Diagonal^2 = 83.0321 + 104.84
Diagonal^2 = 187.8721
Taking the square root of both sides, we get:
Diagonal = sqrt(187.8721)
Diagonal ≈ 13.71 cm
Therefore, the length of the diagonal of the rectangle is approximately 13.71 cm.
To find the length of the diagonal of a rectangle, we first need to find the length and width of the rectangle using the given information.
Let's assume the length of the rectangle is "L" cm.
According to the problem, the width of the rectangle is 8 less than twice its length. So, the width would be 2L - 8 cm.
The area of a rectangle is equal to its length multiplied by its width. So, we have the equation:
Length * Width = 93 cm^2
Substituting the values we found, we get:
L * (2L - 8) = 93
Now, let's solve this equation to find the value of "L" (length of the rectangle).
2L^2 - 8L - 93 = 0
We can solve this quadratic equation using factoring, completing the square, or by using the quadratic formula.
Using the quadratic formula, we have:
L = [ -(-8) ± √((-8)^2 - 4(2)(-93)) ] / (2(2))
L = [ 8 ± √(64 + 744) ] / 4
L = [ 8 ± √808] / 4
L = [8 ± 28.45] / 4
We get two possible values for L:
L₁ = (8 + 28.45) / 4 ≈ 9.86 cm
L₂ = (8 - 28.45) / 4 ≈ -5.89 cm
Since length cannot be negative, we discard the second solution.
Therefore, the length of the rectangle is approximately 9.86 cm.
Now, to find the length of the diagonal, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the other two sides (the length and width of the rectangle).
Using the length and width we found:
Length = 9.86 cm
Width = 2L - 8 = 2(9.86) - 8 = 19.72 - 8 = 11.72 cm
Now, let's calculate the diagonal (D):
D^2 = Length^2 + Width^2
D^2 = 9.86^2 + 11.72^2
D^2 = 97.2996 + 137.4384
D^2 = 234.738
Taking the square root of both sides:
D = √(234.738)
D ≈ 15.31 cm
Therefore, the length of the diagonal of the rectangle is approximately 15.31 cm.