To solve the problem where [tex]\( h \)[/tex] is inversely proportional to the cube of [tex]\( g \)[/tex], we start by understanding what it means for [tex]\( h \)[/tex] to be inversely proportional to [tex]\( g^3 \)[/tex].
This relationship can be written as:
[tex]\[ h \propto \frac{1}{g^3} \][/tex]
To convert the proportionality into an equation, we introduce a constant of proportionality, [tex]\( k \)[/tex]:
[tex]\[ h = \frac{k}{g^3} \][/tex]
Given the values [tex]\( h = 4.5 \)[/tex] when [tex]\( g = 2 \)[/tex], we substitute these into the equation to find [tex]\( k \)[/tex]:
[tex]\[ 4.5 = \frac{k}{2^3} \][/tex]
Next, compute [tex]\( 2^3 \)[/tex]:
[tex]\[ 2^3 = 8 \][/tex]
So the equation becomes:
[tex]\[ 4.5 = \frac{k}{8} \][/tex]
To find [tex]\( k \)[/tex], multiply both sides of the equation by 8:
[tex]\[ k = 4.5 \times 8 \][/tex]
[tex]\[ k = 36.0 \][/tex]
Now that we have determined the constant [tex]\( k \)[/tex], we can rewrite the formula for [tex]\( h \)[/tex] in terms of [tex]\( g \)[/tex]:
[tex]\[ h = \frac{36.0}{g^3} \][/tex]
So the formula for [tex]\( h \)[/tex] in terms of [tex]\( g \)[/tex] is:
[tex]\[ h = \frac{36.0}{g^3} \][/tex]
