Given that [tex]\( v \)[/tex] varies inversely with [tex]\( t \)[/tex], and [tex]\( v = 32 \)[/tex] when [tex]\( t = 4 \)[/tex], what is the value of [tex]\( v \)[/tex] when [tex]\( t = 24 \)[/tex]?
Given that [tex]\( v \)[/tex] varies inversely with [tex]\( t \)[/tex], we can express this relationship as [tex]\( v \propto \frac{1}{t} \)[/tex]. This means that the product of [tex]\( v \)[/tex] and [tex]\( t \)[/tex] is constant. We can write this relationship as:
[tex]\[ v \times t = k \][/tex]
where [tex]\( k \)[/tex] is a constant.
First, we are given that [tex]\( v = 32 \)[/tex] when [tex]\( t = 4 \)[/tex]. We can use these values to find the constant [tex]\( k \)[/tex].
[tex]\[ 32 \times 4 = k \][/tex] [tex]\[ k = 128 \][/tex]
Now, we need to find the value of [tex]\( v \)[/tex] when [tex]\( t = 24 \)[/tex]. Using the constant [tex]\( k \)[/tex] we found, we can set up the equation:
[tex]\[ v \times 24 = 128 \][/tex]
Solving for [tex]\( v \)[/tex]:
[tex]\[ v = \frac{128}{24} \][/tex]
Simplifying the fraction:
[tex]\[ v = 5.333333333333333 \][/tex]
Therefore, the value of [tex]\( v \)[/tex] when [tex]\( t = 24 \)[/tex] is approximately [tex]\( 5.33 \)[/tex].
