How can I write a polynomial equation for this problem and draw a labeled diagram?

A developer wants to build homes on a rectangular plot of land 4 kilometers long and 3 kilometers wide. In this part of the city, regulations require a greenbelt of uniform width along two adjacent sides. The greenbelt must be 10 times the area of the development. Find the width of the greenbelt.

Let the greenbelt width be x. The developed area is (5-x)*(4-x). The greenbelt area is 5x + (4-x)x = 9x - x^2

The requirement is
9x - x^2 = 10 *(x^2 -9x +20)
11 x^2 -99 x +200 = 0
x^2 - 9x + 200/11 = 0

The labeled diagram is one rectangle nested inside a larger triangle with one pair of corners aligned, and a gap of x along the two perpendicular opposite sides

To write a polynomial equation for this problem, we need to define the variables and set up the equation based on the given information.

Let's assume that the width of the greenbelt is represented by variable "x" kilometers.

Now, let's break down the problem and calculate each component:

1. Calculate the area of the rectangular plot of land:
Area = Length * Width
Area = 4 km * 3 km
Area = 12 km²

2. Calculate the area of the greenbelt:
Greenbelt Area = 10 times the area of development = 10 * 12 km²
Greenbelt Area = 120 km²

3. Calculate the length and width of the development area after the greenbelt:
Length of Development = Length - 2x
Width of Development = Width - 2x

4. Calculate the area of the development:
Development Area = (Length of Development) * (Width of Development)
Development Area = (4 km - 2x) * (3 km - 2x)

Since the greenbelt area must be 10 times the area of the development, we can set up the equation:

Greenbelt Area = 120 km² = 10 * Development Area
120 km² = 10 * (4 km - 2x) * (3 km - 2x)

This is a polynomial equation that represents the problem.

To draw a labeled diagram, you can create a rectangle representing the plot of land with dimensions labeled. Then, draw another rectangle inside it, representing the development area. Label the sides and the width of the greenbelt. The width of the greenbelt is represented by the variable "x".