Certainly! To determine the value of the constant [tex]\( k \)[/tex] in the cubic function [tex]\( f(x) = kx^3 \)[/tex] that passes through the point [tex]\((3, 54)\)[/tex], we can follow these detailed steps:
1. We are given the point [tex]\((3, 54)\)[/tex], which means that when [tex]\( x = 3 \)[/tex], the function value [tex]\( f(x) = 54 \)[/tex].
2. Substitute [tex]\( x = 3 \)[/tex] and [tex]\( f(x) = 54 \)[/tex] into the function [tex]\( f(x) = kx^3 \)[/tex].
[tex]\[
f(3) = k \cdot 3^3 = 54
\][/tex]
3. Simplify the equation [tex]\( k \cdot 3^3 = 54 \)[/tex]:
[tex]\[
k \cdot 27 = 54
\][/tex]
4. Solve for [tex]\( k \)[/tex] by isolating it on one side of the equation:
[tex]\[
k = \frac{54}{27}
\][/tex]
5. Perform the division:
[tex]\[
k = 2
\][/tex]
Therefore, the value of the constant [tex]\( k \)[/tex] is [tex]\( 2 \)[/tex].
