Write down an (in)equality which describes the solid ball of radius 2 centered at (-7, -3, 1). It should have a form like (x+15)2+(y-20)2+(z-12)2-100>0,

Question

Write down an (in)equality which describes the solid ball of radius 2 centered at (-7, -3, 1). It should have a form like (x+15)2+(y−20)2+(z−12)2−100>0, where you use one of the following symbols: <,<=,=,>=,>

(We assume that the points on the boundary of the ball belong to the ball.)

The first blank is for the algebraic expression; the second for the (in)equality sign.

Answer 1

The standard form of a sphere in is r² = (x-x₁)²+(y-y₁)²+(z-z₁)²  where (x₁,y_1,z₁) is the center and r is the radius. Here, "sphere" refers to just the surface of a ball like a soccer ball (if air were empty space), and mathematicians use the word "ball" to refer to a filled in sphere, such as an orange.  

 

In order to get the ball, you must include every point in the graph that is a distance from the center less than or equal to the radius, which is where the above answer fails to be correct (the problem statement even says to include the boundary). In other words, the radius must be greater than or equal to the distance of each point in the ball.

 

Therefore r >= sqrt((x-x₁)²+(y-y₁)²+(z-z₁)²), and squaring both sides- r² >= (x-x₁)²+(y-y₁)²+(z-z₁)²

subtracting r², we get the final answer: 0 >= (x+7)²+(y+3)²+(z-1)² -4.

Answer 2

The distance formula in two dimensions is d=√(x₂-x₁)²+(y₂-y₁)² . Using the distance formula, the equation of a circle becomes r² = (x-x₁)² + (y-y₁)² where (x₁,y₁) is the center and the distance from the center to the edge of the circle(the radius) is r.

 

The distance formula and equation of a circle in 3 dimensions is very similar.

d=√(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)² and this nakes the equation of a sphere

r² = (x-x₁)² + (y-y₁)² + (z-z₁)² with (x₁,y₁,z₁) being the center and r being the radius of the sphere.

 

 

this is the skin of the sphere, like a balloon. If you want a solid sphere, you use a less than sign.

So your answer is  (x+7)² + (y+3)² + (z-1)² < 4

 

Hope this helped.