Describing regions: 3 variables

Question

Describe the region bounded by the planes: x=0, y=0, z=0, x+y=4, and x=z-y-1.

It just says to describe the region. 2 things.

1. does anyone know any software that will allow me to draw out these regions.
2. how do i describe this region?

Answer 1

(1) If you are using a Mac, then the application Grapher is really helpful and easy to use (I believe it comes with your Mac, not sure).

(2) There is no unique way to describe it. You can say, for example:

$$ S= (x,y,z) \in \Bbb R^3|x>0 \; , \; y>0 \; , \; z>0 \; , \; z-x-1<y<4-x$$

Answer 2

(see below a picture, done with Matlab).

The 3 first equations show the shape is in the first orthant.

This shape can easily be pictured as a house whose unique room has a right isosceles floor on the ground $z=0$, on which are raised vertical walls :

• $W_1 \perp W_2$ with equations $x=0, \ y=0$ resp. and

• $W_3$ with equation $x+y=4$ making a $45$° angle with $W_1$ and $W_2$.

The last equation describes a roof beginning at height $1$ meter (say) above the origin, reaching five meters above wall $W_3$ because $z=x+y+1$ takes uniform value $5$ when $x+y=4$.

Remark: The slope of the roof is :

$$\text{in radians :} \ \ \arctan(\sqrt{2}) \ \ \approx \ \ 54.7°.$$