Question

application

1. the area of the rectangle shown in the figure is $3x^{2}+38x + 80$. what is the area of the shaded region?

Explanation:

Step1: Find the length of the rectangle

Since the area of a rectangle \(A = l\times w\), and \(A=3x^{2}+38x + 80\), \(w=x + 10\).
We use polynomial long - division or factoring.
Factor \(3x^{2}+38x + 80\):
We need to find two numbers \(m\) and \(n\) such that \(m\times n=3\times80 = 240\) and \(m + n=38\). The numbers are \(30\) and \(8\).
\(3x^{2}+38x + 80=3x^{2}+30x+8x + 80=3x(x + 10)+8(x + 10)=(3x + 8)(x + 10)\)
So the length \(l = 3x+8\)

Step2: Find the length of the shaded region

The non - shaded part has a width of \(x\) (from the figure). The length of the shaded region \(L=(3x + 8)-x=2x + 8\)

Step3: Find the area of the shaded region

The width of the shaded region is \(10\) (from the figure).
Using the area formula \(A=l\times w\), with \(l = 2x+8\) and \(w = 10\)
\(A=10\times(2x + 8)=20x+80\)

Answer:

\(20x + 80\)