To solve the problem, we need to follow the concept of joint variation, which means [tex]\( r \)[/tex] varies jointly as [tex]\( s \)[/tex] and [tex]\( t \)[/tex]. This relationship can be expressed with the equation:
[tex]\[ r = k \cdot s \cdot t \][/tex]
where [tex]\( k \)[/tex] is the constant of variation.
1. Find the constant of variation [tex]\( k \)[/tex]
We're given that [tex]\( r = 8 \)[/tex] when [tex]\( s = 6 \)[/tex] and [tex]\( t = 4 \)[/tex]. Substituting these values into the equation:
[tex]\[
8 = k \cdot 6 \cdot 4
\][/tex]
Simplify:
[tex]\[
8 = k \cdot 24
\][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[
k = \frac{8}{24}
\][/tex]
[tex]\[
k = \frac{1}{3}
\][/tex]
2. Use the constant [tex]\( k \)[/tex] to find the new value of [tex]\( s \)[/tex] when [tex]\( r = 24 \)[/tex] and [tex]\( t = 16 \)[/tex]
Substituting the known values and constant [tex]\( k \)[/tex] into the joint variation equation:
[tex]\[
24 = \frac{1}{3} \cdot s \cdot 16
\][/tex]
Simplify the right-hand side:
[tex]\[
24 = \frac{16}{3} \cdot s
\][/tex]
Multiply both sides by 3 to clear the fraction:
[tex]\[
72 = 16 \cdot s
\][/tex]
Solve for [tex]\( s \)[/tex]:
[tex]\[
s = \frac{72}{16}
\][/tex]
[tex]\[
s = 4.5
\][/tex]
Therefore, when [tex]\( r = 24 \)[/tex] and [tex]\( t = 16 \)[/tex], the value of [tex]\( s \)[/tex] is [tex]\( 4.5 \)[/tex].
