Let's solve this step-by-step and interpret the given functions:
### Step 1: Find the proportionality constant [tex]\( k \)[/tex]
We know that [tex]\( M \)[/tex] is directly proportional to [tex]\( r^3 \)[/tex]. That means we can express it as:
[tex]\[ M = k \cdot r^3 \][/tex]
Given the values [tex]\( r = 4 \)[/tex] and [tex]\( M = 160 \)[/tex], we can substitute these into the equation to find [tex]\( k \)[/tex]:
[tex]\[ 160 = k \cdot (4^3) \][/tex]
[tex]\[ 160 = k \cdot 64 \][/tex]
[tex]\[ k = \frac{160}{64} \][/tex]
[tex]\[ k = 2.5 \][/tex]
### Step 2: Calculate the value of [tex]\( M \)[/tex] when [tex]\( r = 2 \)[/tex]
Now that we have [tex]\( k = 2.5 \)[/tex], we can find [tex]\( M \)[/tex] when [tex]\( r = 2 \)[/tex]. Substitute [tex]\( r = 2 \)[/tex] into the proportionality equation:
[tex]\[ M = 2.5 \cdot (2^3) \][/tex]
[tex]\[ M = 2.5 \cdot 8 \][/tex]
[tex]\[ M = 20.0 \][/tex]
### Step 3: Calculate the value of [tex]\( r \)[/tex] when [tex]\( M = 54 \)[/tex]
Finally, we need to find [tex]\( r \)[/tex] when [tex]\( M = 54 \)[/tex]. Substitute [tex]\( M = 54 \)[/tex] into the proportionality equation:
[tex]\[ 54 = 2.5 \cdot r^3 \][/tex]
[tex]\[ r^3 = \frac{54}{2.5} \][/tex]
[tex]\[ r^3 = 21.6 \][/tex]
[tex]\[ r = \sqrt[3]{21.6} \][/tex]
The cube root of 21.6 is approximately:
[tex]\[ r \approx 2.785 \][/tex]
### Summary of results
1. When [tex]\( r = 2 \)[/tex], [tex]\( M = 20.0 \)[/tex].
2. When [tex]\( M = 54 \)[/tex], [tex]\( r \approx 2.785 \)[/tex].
Thus, these results answer the given problem effectively.
