Question

find the value of x so that the area of the rectangle is equal to the area of the triangle.
image: rectangle with height 8, length ( x + 6 ); triangle with base 12, height ( 3x - 2 )
( x = square )

Explanation:

Step1: Calculate area of rectangle

The formula for the area of a rectangle is \( A_{rectangle} = length \times width \). Here, length is \( x + 6 \) and width is \( 8 \), so \( A_{rectangle}=8(x + 6) \).
Expanding this, we get \( A_{rectangle}=8x + 48 \).

Step2: Calculate area of triangle

The formula for the area of a triangle is \( A_{triangle}=\frac{1}{2}\times base \times height \). Here, base is \( 12 \) and height is \( 3x - 2 \), so \( A_{triangle}=\frac{1}{2}\times12\times(3x - 2) \).
Simplifying, \( \frac{12}{2}=6 \), so \( A_{triangle}=6(3x - 2)=18x - 12 \).

Step3: Set areas equal and solve for x

Since the areas are equal, we set \( 8x + 48 = 18x - 12 \).
Subtract \( 8x \) from both sides: \( 48 = 10x - 12 \).
Add \( 12 \) to both sides: \( 60 = 10x \).
Divide both sides by \( 10 \): \( x = 6 \).

Answer:

\( 6 \)