The problem states that $g = \sqrt{\frac{h-4}{5+h}}$. (a) We need to find the value of $g$ when $h=20$. (b) We need to express $h$ in terms of $g$.

1. Problem Description

The problem states that $g = \sqrt{\frac{h-4}{5+h}}$.

(a) We need to find the value of $g$ when $h=20$.

(b) We need to express $h$ in terms of $g$.

2. Solution Steps

(a) Substituting $h=20$ into the given equation:

$g = \sqrt{\frac{20-4}{5+20}}$

$g = \sqrt{\frac{16}{25}}$

$g = \frac{\sqrt{16}}{\sqrt{25}}$

$g = \frac{4}{5}$

(b) To express $h$ in terms of $g$, we start with the given equation:

$g = \sqrt{\frac{h-4}{5+h}}$

Squaring both sides gives:

$g^2 = \frac{h-4}{5+h}$

Multiplying both sides by $(5+h)$:

$g^2(5+h) = h-4$

$5g^2 + hg^2 = h-4$

Rearrange the equation to isolate $h$:

$hg^2 - h = -4 - 5g^2$

Factor out $h$ on the left side:

$h(g^2 - 1) = -4 - 5g^2$

Divide both sides by $(g^2 - 1)$:

$h = \frac{-4 - 5g^2}{g^2 - 1}$

$h = \frac{-(4 + 5g^2)}{g^2 - 1}$

$h = \frac{4 + 5g^2}{1 - g^2}$

3. Final Answer

(a) $g = \frac{4}{5}$

(b) $h = \frac{4 + 5g^2}{1 - g^2}$