To express the relationship where [tex]\( u \)[/tex] varies jointly as [tex]\( g \)[/tex] and the square of [tex]\( n \)[/tex], we set up the following equation:
[tex]\[ u = k \cdot g \cdot n^2 \][/tex]
Here, [tex]\( k \)[/tex] is the constant of variation.
Now, we solve for [tex]\( g \)[/tex]:
1. Start with the equation:
[tex]\[ u = k \cdot g \cdot n^2 \][/tex]
2. To isolate [tex]\( g \)[/tex], divide both sides of the equation by [tex]\( k \cdot n^2 \)[/tex]:
[tex]\[ g = \frac{u}{k \cdot n^2} \][/tex]
So the equations are:
[tex]\[ u = k \cdot g \cdot n^2 \][/tex]
[tex]\[ g = \frac{u}{k \cdot n^2} \][/tex]
Given specific values, we can now use this formula to calculate [tex]\( g \)[/tex]. For instance, if [tex]\( u = 36 \)[/tex], [tex]\( k = 1 \)[/tex], and [tex]\( n = 3 \)[/tex]:
[tex]\[ g = \frac{36}{1 \cdot 3^2} = \frac{36}{9} = 4.0 \][/tex]
Hence, with the given values, [tex]\( g \)[/tex] is calculated as 4.0.
