To address the question, let's go through Marta's steps to find the value of [tex]\( v \)[/tex] when [tex]\( w = 4 \)[/tex], given [tex]\( v \)[/tex] varies inversely as the cube of [tex]\( w \)[/tex], and [tex]\( w = 6 \)[/tex] and [tex]\( v = 36 \)[/tex]:
Step-by-Step Solution:
1. Understanding the Relationship:
Since [tex]\( v \)[/tex] varies inversely as the cube of [tex]\( w \)[/tex], we can express this relationship with the equation:
[tex]\[
v \cdot w^3 = k
\][/tex]
where [tex]\( k \)[/tex] is a constant.
2. Finding the Constant [tex]\( k \)[/tex]:
We use the given values [tex]\( w = 6 \)[/tex] and [tex]\( v = 36 \)[/tex] to find [tex]\( k \)[/tex].
[tex]\[
36 \cdot 6^3 = k
\][/tex]
Simplifying this:
[tex]\[
36 \cdot 216 = k
\][/tex]
[tex]\[
k = 7776
\][/tex]
3. Finding [tex]\( v \)[/tex] when [tex]\( w = 4 \)[/tex]:
Using the constant [tex]\( k \)[/tex] we found, we substitute [tex]\( w = 4 \)[/tex] into the original equation to find [tex]\( v \)[/tex].
[tex]\[
v \cdot 4^3 = 7776
\][/tex]
Simplifying this:
[tex]\[
v \cdot 64 = 7776
\][/tex]
Solving for [tex]\( v \)[/tex]:
[tex]\[
v = \frac{7776}{64}
\][/tex]
[tex]\[
v = 121.5
\][/tex]
Assessment of Marta's Steps:
Marta's steps showed the following:
1. [tex]\( v \cdot w^3 = k \)[/tex], correct.
2. [tex]\( (36)(6^3) = v(4^3) \)[/tex], correct.
3. [tex]\( (36)(216) = v(64) \)[/tex], correct.
4. [tex]\( 7776 = 64v \)[/tex], correct.
5. [tex]\( 121.5 = v \)[/tex], correct.
Since all of Marta's steps are correct and consistent with the inverse variation relationship and calculations, we can conclude that:
There are no errors in Marta's work.
