Let's solve the problem step-by-step.
Given:
1. The volume [tex]\( v \)[/tex] of a cube varies directly as the cube of the side length [tex]\( r \)[/tex].
2. [tex]\( v = 108 \)[/tex] when [tex]\( x = 3 \)[/tex].
### Part (a): Express [tex]\( v \)[/tex] in terms of [tex]\( x \)[/tex]
Since [tex]\( v \)[/tex] varies directly as the cube of the side length [tex]\( r \)[/tex], we can write the relationship as:
[tex]\[ v = k \cdot r^3 \][/tex]
Where [tex]\( k \)[/tex] is the constant of proportionality. To find [tex]\( k \)[/tex], we use the given values [tex]\( v = 108 \)[/tex] and [tex]\( r = 3 \)[/tex]:
[tex]\[ 108 = k \cdot 3^3 \][/tex]
[tex]\[ 108 = k \cdot 27 \][/tex]
[tex]\[ k = \frac{108}{27} \][/tex]
[tex]\[ k = 4 \][/tex]
Now that we know [tex]\( k = 4 \)[/tex], the equation can be written as:
[tex]\[ v = 4x^3 \][/tex]
### Part (b): Find the volume of the cube when [tex]\( x = 6 \)[/tex]
Using the equation [tex]\( v = 4x^3 \)[/tex]:
[tex]\[ v = 4 \cdot 6^3 \][/tex]
[tex]\[ v = 4 \cdot 216 \][/tex]
[tex]\[ v = 864 \][/tex]
So, the volume of the cube when [tex]\( x = 6 \)[/tex] is [tex]\( 864 \)[/tex].
### Part (c): Find the length of side [tex]\( x \)[/tex] when [tex]\( v = 4,000 \)[/tex]
We use the equation [tex]\( v = 4x^3 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 4000 = 4x^3 \][/tex]
[tex]\[ x^3 = \frac{4000}{4} \][/tex]
[tex]\[ x^3 = 1000 \][/tex]
[tex]\[ x = \sqrt[3]{1000} \][/tex]
[tex]\[ x = 10 \][/tex]
So, the length of side [tex]\( x \)[/tex] when [tex]\( v = 4,000 \)[/tex] is [tex]\( 10 \)[/tex].
### Summary
a) The volume [tex]\( v \)[/tex] in terms of [tex]\( x \)[/tex] is given by [tex]\( v = 4x^3 \)[/tex].
b) The volume of the cube when [tex]\( x = 6 \)[/tex] is [tex]\( 864 \)[/tex].
c) The length of side [tex]\( x \)[/tex] if [tex]\( v = 4,000 \)[/tex] is [tex]\( 10 \)[/tex].
By following these steps, we've solved each part of the problem using the relationships and principles given.
