Given that \( H \) is inversely proportional to the cube root of \( L \), we can express this relationship mathematically as:
[tex]\[ H = \frac{k}{\sqrt[3]{L}} \][/tex]
where \( k \) is a constant of proportionality.
We are also given that \( H = 7 \) when \( L = 64 \).
1. Substitute \( H = 7 \) and \( L = 64 \) into the equation to find \( k \):
[tex]\[ 7 = \frac{k}{\sqrt[3]{64}} \][/tex]
2. Calculate the cube root of \( 64 \):
[tex]\[ \sqrt[3]{64} = 4 \][/tex]
3. Substitute \(\sqrt[3]{64} = 4\) into the equation:
[tex]\[ 7 = \frac{k}{4} \][/tex]
4. Solve for \( k \):
[tex]\[ k = 7 \times 4 \][/tex]
[tex]\[ k = 28 \][/tex]
Thus, the constant \( k \) is \( 28 \).
5. Substitute \( k = 28 \) back into the original equation:
[tex]\[ H = \frac{28}{\sqrt[3]{L}} \][/tex]
Therefore, the equation connecting \( H \) and \( L \) is:
[tex]\[ H = \frac{28}{\sqrt[3]{L}} \][/tex]
