Question

1. a rectangle with a length of 20 centimeters and an unknown width of x centimeters has a smaller rectangle cut out of it. the smaller rectangle has a length of 16 cm and a width of x - 3. the area of the shaded region (what remains) is 80 square centimeters. set up and solve an equation for the value of x.

Explanation:

Step1: Find area of large rectangle

The area of a rectangle is $A = length\times width$. For the large rectangle, with length $l = 20$ cm and width $w=x$ cm, its area $A_{1}=20x$ square - centimeters.

Step2: Find area of small rectangle

For the small rectangle, with length $l = 16$ cm and width $w=(x - 3)$ cm, its area $A_{2}=16(x - 3)$ square - centimeters.

Step3: Set up the equation

The area of the shaded region is the area of the large rectangle minus the area of the small rectangle. So, $A_{1}-A_{2}=80$. Substitute the expressions for $A_{1}$ and $A_{2}$:
\[

$$\begin{align*} 20x-16(x - 3)&=80\\ 20x-16x+48&=80\\ 4x+48&=80 \end{align*}$$

\]

Step4: Solve the equation for $x$

Subtract 48 from both sides of the equation: $4x+48 - 48=80 - 48$, which gives $4x = 32$. Then divide both sides by 4: $\frac{4x}{4}=\frac{32}{4}$, so $x = 8$.

Answer:

$x = 8$