Question

The length of a rectangle is 5 5 centimeters less than six times six times its width. Its area is 14 14 square centimeters. Find the dimensions of the rectangle.

Answer 1

Dimensions of rectangle are length = 7 cm and width = 2 cm

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,

`x=(-b\pm √(b^2-4ac))/(2a)`

Rewriting the formula in terms of w,

`w=(-b\pm √(b^2-4ac))/(2a)`

From the 6 w² - 5 w - 14 = 0, value of a = 6 , b = - 5 and c = - 14.

Substituting the values,

`w=(-\left(-5\right)\pm √(\left(-5\right)^2-4\cdot \:6\left(-14\right)))/(2\cdot \:6)`

Simplifying,

`w=(-\left(-5\right)\pm √(25+336))/(12)`

`w=(-\left(-5\right)\pm √(361))/(12)`

Since √361 = 19,

`w=(-\left(-5\right)\pm 19)/(12)`

There will be two values of w,

`w=(-\left(-5\right)+19)/(12)` and
`w=(-\left(-5\right)-19)/(12)`

Simplifying,

`w=(5+19)/(12)` and
`w=(5-19)/(12)`

`w=(24)/(12)` and
`w=(-14)/(12)`

Reducing the fraction in lowest form, divide first expression by 12 and second expression by 2.

`w=2` and
`w=(-7)/(6)`

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.