Part Variation - Questions
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The variable \(S\) is the sum of two quantities, one of which varies with \(x\) and the other as the square of \(y\). If \(S = 20\) when \(x = 4\) and \(y = 2\), and \(S = 13\) when \(x = 5\) and \(y = 1\), find \(S\) when \(x = 3\) and \(y = 4.\)
$$
\begin{aligned}
S &\propto x+y^{2}\\
S&=k_{1} x+k_{2} y^{2}\\
20&=4 k_{1}+4 k_{2}\\
5&=k_{1}+k_{2} \color{red}\cdots \text { (1) }\\
13&=5 k_{1}+k_{2} \color{red}\cdots \text { (2) }\\
{\color{red}(2)}-{\color{red}(1)} \quad 4 k_{1}&=8\\
k_{1}&=2\\
\text{in}\ {\color{red}(1)} \quad 5&=2+k_{2}\\
k_{2}&=3\\
S &=2 x+3 y^{2} \\
x&=3, y=4 \\
S &=6+3 \times 4^{2} \\
&=6+48 \\
S &=54
\end{aligned}
$$
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