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If r varies inversely as the square root of s and t, how does s vary with r and ...


Question

If r varies inversely as the square root of s and t, how does s vary with r and t?

Options

A) s varies inversely as r and t2

B) s varies inverely as r2 and t

C) s varies directly as r2 and t2

D) s varies directly as r and t

The correct answer is B.

Explanation:

\(r \propto \frac{1}{\sqrt{s}}, r \propto \frac{1}{\sqrt{t}}\)
\(r \propto \frac{1}{\sqrt{s}}\) ..... (1)
\(r \propto \frac{1}{\sqrt{t}}\) ..... (2)
Combining (1) and (2), we get
\(r = \frac{k}{\sqrt{s} \times \sqrt{t}} = \frac{k}{\sqrt{st}}\)
This gives \(\sqrt{st} = \frac{k}{r}\)
By taking the square of both sides, we get
st = \(\frac{k^2}{r^2}\)
s = \(\frac{k^2}{r^{2}t}\)

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Dicussion (1)

  • \(r \propto \frac{1}{\sqrt{s}}, r \propto \frac{1}{\sqrt{t}}\)
    \(r \propto \frac{1}{\sqrt{s}}\) ..... (1)
    \(r \propto \frac{1}{\sqrt{t}}\) ..... (2)
    Combining (1) and (2), we get
    \(r = \frac{k}{\sqrt{s} \times \sqrt{t}} = \frac{k}{\sqrt{st}}\)
    This gives \(\sqrt{st} = \frac{k}{r}\)
    By taking the square of both sides, we get
    st = \(\frac{k^2}{r^2}\)
    s = \(\frac{k^2}{r^{2}t}\)

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