To solve the problem, we start with the direct variation relationship, which can be expressed mathematically as:
\[ S = k T^{4/5} \]
where \( k \) is the constant of proportionality.
From the problem, we know that \( S = 48 \) when \( T = 32 \). We can substitute these values into the equation to find \( k \):
\[ 48 = k (32^{4/5}) \]
Now, we calculate \( 32^{4/5} \):
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Calculate \( 32^{1/5} \): \[ 32^{1/5} = 2 \quad (\text{since } 2^5 = 32) \]
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Now calculate \( 32^{4/5} \): \[ 32^{4/5} = (32^{1/5})^4 = 2^4 = 16 \]
Now substitute this back into the equation to solve for \( k \):
\[ 48 = k (16) \]
To find \( k \): \[ k = \frac{48}{16} = 3 \]
Now we have the direct variation equation:
\[ S = 3 T^{4/5} \]
Next, we want to find \( S \) when \( T = 243 \):
\[ S = 3 (243^{4/5}) \]
To calculate \( 243^{4/5} \):
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Calculate \( 243^{1/5} \): \[ 243^{1/5} = 3 \quad (\text{since } 3^5 = 243) \]
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Now calculate \( 243^{4/5} \): \[ 243^{4/5} = (243^{1/5})^4 = 3^4 = 81 \]
Now substitute this back into the equation for \( S \):
\[ S = 3 \times 81 = 243 \]
Thus, when \( T = 243 \):
\[ S = 243 \]
So the final answer is:
\[ \boxed{243} \]
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