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