Question

1. a rectangle has a length of (4 + x) inches and a width of 12 inches. a triangle has a base of 24 inches and a height of (2x + 6) inches. the area of the rectangle is equal to the area of the triangle. what is the value of x?

Explanation:

Step1: Write area formulas

The area of a rectangle $A_{r}=l\times w$, where $l = 4 + x$ and $w = 12$, so $A_{r}=12(4 + x)$. The area of a triangle $A_{t}=\frac{1}{2}\times b\times h$, where $b = 24$ and $h=2x + 6$, so $A_{t}=\frac{1}{2}\times24\times(2x + 6)$.

Step2: Set up the equation

Since $A_{r}=A_{t}$, we have $12(4 + x)=\frac{1}{2}\times24\times(2x + 6)$.

Step3: Simplify the equation

First, simplify the right - hand side: $\frac{1}{2}\times24\times(2x + 6)=12(2x + 6)$. The equation becomes $12(4 + x)=12(2x + 6)$. Divide both sides by 12, we get $4 + x=2x+6$.

Step4: Solve for x

Subtract $x$ from both sides: $4=2x - x+6$, which simplifies to $4=x + 6$. Then subtract 6 from both sides: $x=4 - 6=-2$.

Answer:

$x=-2$