Question

1. spicy! a rectangle has a length of (x) inches and a width of (x + 11) inches. a triangle has a base of (2x + 6) inches and a height of (x + 5) inches. the area in square inches of the rectangle is equal to the area in square inches of the triangle. what is the value of x? show all of your work in a clear and organized manner. recall: for a triangle, $a = \frac{bh}{2}$ or $a = \frac{1}{2}bh$ a) find the area of the triangle. b) find the area of the rectangle. c) the area in square inches of the rectangle is equal to the area in square inches of the triangle. what is the value of x?

Explanation:

Step1: Calculate triangle area

Use formula $A=\frac{1}{2}bh$. Substitute $b=2x+6$, $h=x+5$:

$$\begin{align*} A_{\text{triangle}}&=\frac{1}{2}(2x+6)(x+5)\\ &=(x+3)(x+5)\\ &=x^2+8x+15 \end{align*}$$

Step2: Calculate rectangle area

Use formula $A=lw$. Substitute $l=x$, $w=x+11$:

$$\begin{align*} A_{\text{rectangle}}&=x(x+11)\\ &=x^2+11x \end{align*}$$

Step3: Set areas equal, solve for $x$

Set $x^2+11x=x^2+8x+15$:

$$\begin{align*} x^2+11x-x^2-8x&=15\\ 3x&=15\\ x&=5 \end{align*}$$

Answer:

a) $x^2+8x+15$
b) $x^2+11x$
c) $5$