Sure! Let's go through the problem step-by-step.
1. Understanding the Relationship:
The variable [tex]\( m \)[/tex] varies directly as the cube root of [tex]\( n \)[/tex]. This means there is a constant [tex]\( k \)[/tex] such that:
[tex]\[
m = k \sqrt[3]{n}
\][/tex]
2. Finding the Constant [tex]\( k \)[/tex]:
We are given that when [tex]\( n = 27 \)[/tex], [tex]\( m = 216 \)[/tex]. Using this, we can find [tex]\( k \)[/tex]:
[tex]\[
216 = k \sqrt[3]{27}
\][/tex]
The cube root of 27 is 3:
[tex]\[
216 = k \times 3
\][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[
k = \frac{216}{3} = 72
\][/tex]
3. Constructing the Equation:
Now that we have [tex]\( k \)[/tex], we can write the equation for [tex]\( m \)[/tex] and [tex]\( n \)[/tex] as:
[tex]\[
m = 72 \sqrt[3]{n}
\][/tex]
4. Checking the Options:
a) [tex]\( m = 8n \)[/tex]
- Substituting [tex]\( n = 27 \)[/tex]:
[tex]\[
m = 8 \times 27 = 216
\][/tex]
Verified as true for this specific case, but it doesn’t represent the direct proportionality to the cube root of [tex]\( n \)[/tex].
b) [tex]\( m = 72 \sqrt[3]{n} \)[/tex]
- This is the equation we derived, and it matches perfectly.
c) [tex]\( m \sqrt[3]{n} = 648 \)[/tex]
- Substitute [tex]\( m = 216 \)[/tex] and [tex]\( \sqrt[3]{n} = 3 \)[/tex] (since [tex]\( n = 27 \)[/tex]):
[tex]\[
216 \times 3 = 648
\][/tex]
True, but it's another representation rather than the original form.
d) [tex]\( m n = 5,832 \)[/tex]
- Substitute [tex]\( m = 216 \)[/tex] and [tex]\( n = 27 \)[/tex]:
[tex]\[
216 \times 27 = 5832
\][/tex]
True for the given values, but not the direct relationship to the cube root of [tex]\( n \)[/tex].
Given all options, the one that reliably describes the relationship between [tex]\( m \)[/tex] and [tex]\( n \)[/tex] is:
Answer: B. [tex]\( m = 72 \sqrt[3]{n} \)[/tex]
