**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 t^{2}

B) s varies inverely as r^{2} and t

C) s varies directly as r^{2} and t^{2}

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 = \(\frac{k^2}{r^{2}t}\)

\(r \propto \frac{1}{\sqrt{s}}\) ..... (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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\(r \propto \frac{1}{\sqrt{s}}, r \propto \frac{1}{\sqrt{t}}\)

\(r \propto \frac{1}{\sqrt{s}}\) ..... (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 = \(\frac{k^2}{r^{2}t}\)