Answer :
Answer:
The length of the diagonal = 3.6 feet
Step-by-step explanation:
* Lets change the story problem to equation to solve it
- Diagonal in a rectangle is the line which joining two
opposite vertices in it
- The diagonal , the length and the width formed together right
angle triangle where the diagonal is the hypotenuse of the
triangle and the length, the width are the two legs of the
right angle
- The length of the diagonal d is √(l² + w²) ⇒ Pythagoras theorem
∴ d² = l² + w²
- The diagonal is 2 feet more than the width
∴ d = w + 2
- The length is twice the width
∴ l = w
* Lets substitute the value of d and l in the equation of Pythagoras
∴ (w + 2)² = (2w)² + w² ⇒ solve the bracket
∴ w² + 4w + 4 = 4w² + w² ⇒ collect them in one side and add like term
∴ 4w² + w² - w² - 4w - 4 = 0 ⇒ add the like term
∴ 4w² - 4w - 4 = 0 ⇒ divide all term by 4
∴ w² - w - 1 = 0 ⇒ solve the quadratic using the formula
- In ax² + bx + c = 0
[tex]x=\frac{-b+/-\sqrt{b^{2}-4ac}}{2a}[/tex]
∵ a = 1 , b = -1 , c = -1
∴ [tex]x=\frac{-(-1)+\sqrt{(-1)^{2}-4(1)(-1)}}{2(1)}=\frac{1+\sqrt{5}}{2}=1.6feet[/tex]
-The value of the width is 1.6 feet
∴ The length = 1.6 × 2 = 3.2 feet
∴ The length of the diagonal = 2 + 1.6 = 3.6 feet
Rectangle has its adjacent sides perpendicular to each other. The length of the plywood's diagonal is approx 3.62 feet.
What is Pythagoras Theorem?
If ABC is a triangle with AC as the hypotenuse and angle B with 90 degrees then we have:
[tex]|AC|^2 = |AB|^2 + |BC|^2[/tex]
where |AB| = length of line segment AB.
What are the roots of a quadratic equation?
If the quadratic equation is of the form [tex]ax^2 + bx + c = 0[/tex] for constants a, b, and c, and for variable x, then its roots are given as:
[tex]x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
In a rectangle, its adjacent sides are perpendicular to each other(making 90 degrees with each other), thus, when its diagonal is drawn, we can use Pythagoras theorem to relate its length, width and length of the diagonals (diagonal as hypotenuse) (both of diagonals of a rectangle are of same length).
Thus, using Pythagoras theorem, we get:
[tex]|Diagonal|^2 = |Length|^2 + |Width|^2[/tex] (for a rectangle).
For the given case, let we assume:
- Length of the plywood = L
- Width of the plywood = W
- Diagonal of the plywood = D
Then, [tex]D^2 = L^2 + W^2\\[/tex]
From the given facts, we get:
D = 2 + W (diagonal which measures two feet more than the width)
and L = 2W (length of the plywood is twice the width)
Expressing L and W in terms of D(since we need D's value), we get:
W = D - 2
and L = 2W = 2(D-2)
Putting these values in the equation we got from Pythagoras theorem,
[tex]D^2= L^2 + W^2\\D^2 = (2(D-2))^2 + (D-2)^2\\D^2 = 4D^2 + 16 - 16D + D^2 + 4 - 4D\\4D^2 -20D + 20 = 0\\D^2 -5D + 5 = 0[/tex]
Using the root formula for quadratic equation, we get:
[tex]D = \dfrac{5 \pm \sqrt{(-5)^2 -20}}{2} \approx \dfrac{5 \pm 2.24}{2}\\\\D \approx 1.38, 3.62\\[/tex]
Since W = D - 2, so D cannot be 1.38 feet since for that, width will go negative which isn't allowed as width is length which cannot be non-negative.
Thus, the length of the considered plywood's diagonal is 3.62 feet approx.
Learn more about solutions of quadratic equations here:
brainly.com/question/3358603