To determine the value of [tex]\( z \)[/tex] when [tex]\( x = 4 \)[/tex] and [tex]\( y = 9 \)[/tex], following the given conditions that [tex]\( z \)[/tex] varies directly with [tex]\( x^2 \)[/tex] and inversely with [tex]\( y \)[/tex], we can outline each step of the solution using the described variation relationship.
First, we interpret the relationship [tex]\( z \)[/tex] varies directly with [tex]\( x^2 \)[/tex] and inversely with [tex]\( y \)[/tex]. This means we can write the equation in the form:
[tex]\[ z = k \cdot \frac{x^2}{y} \][/tex]
where [tex]\( k \)[/tex] is a constant.
Given the initial conditions:
[tex]\[ x_1 = 2, \quad y_1 = 4, \quad z_1 = 3 \][/tex]
We substitute these values into our equation to find the constant [tex]\( k \)[/tex]:
[tex]\[ z_1 = k \cdot \frac{x_1^2}{y_1} \][/tex]
[tex]\[ 3 = k \cdot \frac{2^2}{4} \][/tex]
[tex]\[ 3 = k \cdot \frac{4}{4} \][/tex]
[tex]\[ 3 = k \cdot 1 \][/tex]
[tex]\[ k = 3 \][/tex]
Now, we use this constant [tex]\( k \)[/tex] to find the new value of [tex]\( z \)[/tex] when [tex]\( x_2 = 4 \)[/tex] and [tex]\( y_2 = 9 \)[/tex]:
[tex]\[ z_2 = k \cdot \frac{x_2^2}{y_2} \][/tex]
[tex]\[ z_2 = 3 \cdot \frac{4^2}{9} \][/tex]
[tex]\[ z_2 = 3 \cdot \frac{16}{9} \][/tex]
[tex]\[ z_2 = \frac{48}{9} \][/tex]
[tex]\[ z_2 = \frac{16}{3} \][/tex]
Therefore, the value of [tex]\( z \)[/tex] when [tex]\( x = 4 \)[/tex] and [tex]\( y = 9 \)[/tex] is:
[tex]\[ \boxed{\frac{16}{3}} \][/tex]
