If \( a \) varies inversely as \( b^2 \), we can write the relationship as:
\[ a = \frac{k}{b^2} \]
where \( k \) is a constant.
Given that \( a = 48 \) when \( b = 4 \), we can substitute these values into the equation to find \( k \):
\[ 48 = \frac{k}{4^2} \]
This simplifies to:
\[ 48 = \frac{k}{16} \]
Now, multiply both sides by \( 16 \):
\[ k = 48 \times 16 \]
Calculating \( 48 \times 16 \):
\[ k = 768 \]
Now we have the relationship:
\[ a = \frac{768}{b^2} \]
Next, we need to find \( a \) when \( b = 7 \):
\[ a = \frac{768}{7^2} \]
Calculating \( 7^2 \):
\[ 7^2 = 49 \]
Now substitute this back into the equation:
\[ a = \frac{768}{49} \]
Now, dividing \( 768 \) by \( 49 \):
\[ a \approx 15.6734693877551 \]
Thus, the value of \( a \) when \( b = 7 \) is approximately:
\[ a \approx 15.67 \]
To summarize:
\[ \text{When } b = 7, , a \approx 15.67. \]