Respuesta :
Answer:
Dimensions of rectangle are length = 7 cm and width = 2 cm
Step-by-step explanation:
Let l = length and w = width of the rectangle.
Given that, length is 5 cm less than 6 times width. Rewriting it in equation form,
L = 6 w - 5 cm
Also given that area of rectangle is 14 cm²
Now using formula for area of rectangle,
Area of rectangle = length × width
Area of rectangle = l × w
Substituting the values,
14 = (6 w - 5) × w
Simplifying by using distributive rule,
14 = 6 w² - 5 w
To find the value of b, use the quadratic formula. So rewriting the equation in quadratic form ax²+bx+c=0
Subtracting 14 on both sides,
0 = 6 w² - 5 w - 14
Rewriting,
6 w² - 5 w - 14 = 0
Now applying quadratic formula,
[tex]x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Rewriting the formula in terms of w,
[tex]w=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
From the 6 w² - 5 w - 14 = 0, value of a = 6 , b = - 5 and c = - 14.
Substituting the values,
[tex]w=\dfrac{-\left(-5\right)\pm \sqrt{\left(-5\right)^2-4\cdot \:6\left(-14\right)}}{2\cdot \:6}[/tex]
Simplifying,
[tex]w=\dfrac{-\left(-5\right)\pm \sqrt{25+336}}{12}[/tex]
[tex]w=\dfrac{-\left(-5\right)\pm \sqrt{361}}{12}[/tex]
Since √361 = 19,
[tex]w=\dfrac{-\left(-5\right)\pm 19}{12}[/tex]
There will be two values of w,
[tex]w=\dfrac{-\left(-5\right)+19}{12}[/tex] and [tex]w=\dfrac{-\left(-5\right)-19}{12}[/tex]
Simplifying,
[tex]w=\dfrac{5+19}{12}[/tex] and [tex]w=\dfrac{5-19}{12}[/tex]
[tex]w=\dfrac{24}{12}[/tex] and [tex]w=\dfrac{-14}{12}[/tex]
Reducing the fraction in lowest form, divide first expression by 12 and second expression by 2.
[tex]w=2[/tex] and [tex]w=\dfrac{-7}{6}[/tex]
Since the length of width cannot be negative, so value of w = 2 cm
To find the value of length, Use equation L = 6 w - 5 cm
L = 6 (2) - 5 cm
L = 12 - 5 cm
L = 7 cm
Therefore l = 7 cm and w = 2 cm are the dimensions of the rectangle.
