Find the dimensions of the rectangle

Question

The area of a rectangle is 45 yd² the length of the rectangle is 1 yd more than twice the width. Find the dimensions of the rectangle

Answer 1

We'll call the length, x, and the width, y.

 

We know the length is 1 yd more than twice the width:

2y + 1 =x

 

 

We also know that the area is 45 yd² and that the area of the rectangle is length * width:

x * y = 45

(2y + 1) * y = 45

2y² + y=45

2y² + y -45 = 0

(2y-9)(y+5) = 0 ---> 2y = 9; y = 4.5 OR y = -5

We know that lengths and widths have to have a positive value, so we'll choose y = 4.5

 

Now find x:

x = 2y + 1

x = 9 + 1 = 10

 

 

Let's check our answer: length * width= 10 * 4.5 =45

 

Answer 2

Let width = x

Let length = 2x + 1

Area = 45 yd²

 

The area of a rectangle is

 

length × width

 

 

x(2x + 1) = 45

 

2x² + x = 45

 

 

Subtract 45 on both sides of equation.

 

 

2x² + x - 45 = 0

 

(2x - 9)(x + 5) = 0

 

 

Set the factors equal to zero.

 

2x - 9 = 0          and          x + 5  = 0

2x = 9               and          x = -5

 

x = 4.5

 

We accept the positive value of x since one of the dimensions cannot be a negative value.

 

width = 4.5

length = 2(4.5) + 1 = 9 + 1 = 10

 

The width is 4.5 yds and the length is 10 yds.

 

Answer 3

Let l = length and w = width.

 

From the problem, we know that "the length of the rectangle is 1 yd more than twice the width" which means:

 

"l is 1 yd more than twice w"

"l is 1 yd more than 2w"

"l is 2w + 1"

l = 2w + 1

 

The equation for the area (A) of a rectangle is A = l×w.

 

Let's plug in A = 45 and l = 2w + 1:

 

45 = (2w + 1)×w

 

Now solve for w:

 

45 = w(2w + 1)

45 = 2w² + w

0 = 2w² + w - 45

 

This is a quadratic equation, which factors as 0 = (2w - 9)(w + 5).

 

That means that:

 

2w - 9 = 0

2w = 9

w = 4.5

 

OR

 

w + 5 = 0

w = -5

 

We can reject -5, since there's no such thing as a negative yard. So, width = 4.5 yards.

 

Finally, to find length, plug w = 4.5 into l = 2w + 1:

 

l = 2(4.5) + 1

l = 9 + 1

l = 10

 

So, length = 10 yards.