Solution -
The sphere is centered in the cube, so diameter of the sphere is equal to the side of the cube. Hence ,the radius of the sphere is half of the length of the side.
If the length of side of the cibe = a , then radius of the sphere is
Volume of the cube = a³, volume of the sphere = π(radius)³ =
π(
)³
volume of the cube outside the sphere = volume of the cube - volume of the sphere
= π(
)³
=
=a³(1 - )
The distance between two real numbers is the absolute value of their difference,
. So the statement "all
that are 1/4 away from -6" is captured in the equation
or
To solve, you can use the definition of absolute value: if is not negative, then
; otherwise,
.
So there are two cases to consider:
1. Suppose . Then
, and the equation reduces to
2. Suppose . Then
and
, so the two solutions we found are correct because both
and
are indeed 1/4 away from -6.
- - -
For the question regarding inequality, you know that the absolute value must return a non-negative number. In other words, there is no such that
.