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

Answer:

$\huge\boxed{x = 1 \ \ \ \ OR \ \ \ \ x = -\frac{10}{3} }$

Step-by-step 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

Answer 2

Answer:

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:

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

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

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

$3x^2+7x-10=0\\\\x_1=\frac{-10}{3}\\x_2 = 1$

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