Question

The area of a rectangle is 65ft∧2 , and the length of the rectangle is 3ft less than twice the width. Find the dimensions of the rectangle.

Answer 1

Width = 6.5 ft

Length = 10 ft

Explanation:

• We can allow the variables w and l to represent the width and length of the rectangle.

• The formula for the area of a rectangle is given by:

A = lw, where

• A is the area in units squared,
• l is the length,
• and w is the width.

Since the area of the rectangle is 65 ft^2, so we can use the following equation to represent the are:

l * w = 65

Since we're also told that the length is 3 ft less than twice the width, we can represent this with the following equation:

l = 2w - 3.

Substituting the expression for l from the second equation into the first equation gives us:

(2w - 3) * w = 65

Expanding this expression gives us:

2w^2 - 3w = 65

Subtracting 65 from both sides gives us:

2w^2 - 3w - 65 = 0

We can solve this quadratic equation for w using the quadratic formula:

`w=(-b+/-√(b^2-4ac) )/(2a)`

Plugging these values into the formula gives us:

`w=(-(-3)+/-√((-3)^2-4(2)(-65)) )/(2(2)) \\\\w=((3)+/-√((9+520)) )/(4) \\\\w=((3)+/-√((529)) )/(4)\\\\w=3/4+(1/4*√(529))=6.5\\ w=3/4-(1/4*√(529))=-5`

We can't have a negative dimension so the width is 6.5 ft.

Now we can plug in 6.5 for w and 65 for A in the rectangle area formula to find l, the length of the rectangle:

(65 = 6.5l) / 6.5

10 = l

Thus, the length is 10 ft.

Answer 2

Let w = width

Then length = 2w - 3

Area is found by multiplying width x length

w(2w - 3) = 65

2w^2 - 3w - 65 = 0

(2w -13)(w + 5) = 0

2w -13 = 0 Or w + 5 = 0
2w = 13. w = -5
w = 6.5

Width is 6.5 ft and Length is 10 ft