Question

Given that a rectangle with length $3x$ inches and width $x + 5$ inches has the property that its area and perimeter have equal values, what is $x$

Answer 1

First, we need to know how to calculate the area and the permiter of a rectangle.

To calculate the area, we multiply base by height and to calculate the perimeter, we sum all sides.

Knowing this, we can say that the area is 3x * (x+5) and the perimiter is 3x + 3x + x + 5 + x + 5, as we know both are the same, we write it as an equation:

`3x * (x+5) = 3x + 3x + x + 5 + x + 5`

Now we solve the equation:

`3x^2 +15x = 6x + 2x + 10`

`3x^2+15x =8x + 10`

`3x^2+15x-8x-10=0`

`3x^2+7x-10=0\\\\x_1=(-10)/(3)\\x_2 = 1`

As the negative result doesn't have sense, we only pick the second one: 1.

If x = 1, then area would be 3*6 = 18 square inches and perimeter 3+3+6+6 = 18 inches

Answer 2

`\huge\boxed{x = 1 \ \ \ \ OR \ \ \ \ x = -(10)/(3) }`

Explanation:

Length = 3x

Width = x + 5

Area of Rectangle:

=> (Length)(Width)

=> (3x)(x+5)

=>
`3x^2 + 15x`

Perimeter of Rectangle:

=> 2 (Length) + 2 (Width)

=> 2(3x) + 2(x+5)

=> 6x + 2x + 10

=> 8x + 10

Given Condition is:

Perimeter = Area

`3x^2 + 15x` = 8x + 10

`3x^2 + 15 x -8x - 10 = 0\\3x^2 + 7x -10 = 0\\3x^2 + 10 x - 3x -10 = 0\\x(3x+10) - 1(3x+10) = 0\\(x-1) (3x+10) = 0`

Either,

x - 1 = 0 OR 3x + 10 = 0

x = 1 OR 3x = -10

x = 1 OR x = -10 / 3