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Suppose [tex]\( z \)[/tex] varies inversely with [tex]\( t \)[/tex] and that [tex]\( z = 9 \)[/tex] when [tex]\( t = 11 \)[/tex]. What is the value of [tex]\( z \)[/tex] when [tex]\( t = 3 \)[/tex]?
When a variable [tex]\( z \)[/tex] varies inversely with another variable [tex]\( t \)[/tex], the relationship between them can be described by the equation:
[tex]\[ z \times t = k \][/tex]
where [tex]\( k \)[/tex] is a constant. Given that [tex]\( z = 9 \)[/tex] when [tex]\( t = 11 \)[/tex], we can determine the constant [tex]\( k \)[/tex] by plugging in these values:
[tex]\[ 9 \times 11 = k \][/tex] [tex]\[ k = 99 \][/tex]
Now that we have the constant [tex]\( k \)[/tex], we can use it to find the value of [tex]\( z \)[/tex] for any other value of [tex]\( t \)[/tex].
We need to find [tex]\( z \)[/tex] when [tex]\( t = 3 \)[/tex]. We start by setting up the inverse variation equation with the known constant [tex]\( k \)[/tex]:
[tex]\[ z \times 3 = 99 \][/tex]
To solve for [tex]\( z \)[/tex], we divide both sides of the equation by 3:
[tex]\[ z = \frac{99}{3} \][/tex] [tex]\[ z = 33 \][/tex]
Therefore, the value of [tex]\( z \)[/tex] when [tex]\( t = 3 \)[/tex] is [tex]\( 33 \)[/tex].
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