Question

riley wants to put a poster in a frame. the original poster is 11 inches wide and 14 in figure shows the poster and its frame. riley knows he wants the total area of the poster and frame together to be no more than 320 which inequality correctly models this situation? a. $x^{2}+11x + 14leq320$ b. $x^{2}+25x + 154leq320$ c. $4x^{2}+11x + 14leq320$ d. $4x^{2}+50x + 154leq320$

Explanation:

Step1: Determine dimensions of frame + poster

The frame has a width of \( x \) on both sides, so the total width of the poster and frame is \( 11 + 2x \) inches. The total height is \( 14 + 2x \) inches.

Step2: Calculate the area

The area \( A \) of a rectangle is length times width, so \( A=(11 + 2x)(14 + 2x) \).
Expand the product:
\[

$$\begin{align*} (11 + 2x)(14 + 2x)&=11\times14+11\times2x+2x\times14 + 2x\times2x\\ &=154 + 22x+28x + 4x^{2}\\ &=4x^{2}+50x + 154 \end{align*}$$

\]
Since the total area should be no more than 320, the inequality is \( 4x^{2}+50x + 154\leq320 \).

Answer:

D. \( 4x^{2} + 50x + 154 \leq 320 \)