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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

### 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 = $$\frac{k^2}{r^{2}t}$$

## 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 = $$\frac{k^2}{r^{2}t}$$