液体以 15 cm$^3$/s 的恒定速率注入一个容器中。当容器内液体的深度为 $x$ cm 时，容器内液体的体积 $V$ cm$^3$ 由下式给出：
$V = \frac{1}{4}x(x + 16).$ 当 $V = 20$ 时，求：

1. Substitute $V = 20$ into the given equation and rearrange into standard quadratic form

$20 = \frac{1}{4}x(x + 16)$
$80 = x^2 + 16x$
$x^2 + 16x - 80 = 0$

2. Solve the quadratic equation and reject the invalid root

$(x + 20)(x - 4) = 0$
$x = -20 \text{ or } x = 4$
Since depth $x$ must be positive, $x = 4$.

1. Differentiate $V$ with respect to $x$

$V = \frac{1}{4}x^2 + 4x$

$\frac{dV}{dx} = \frac{1}{2}x + 4$

2. Evaluate $\frac{dV}{dx}$ at $x = 4$ (from part i)

When $x = 4$,
$\frac{dV}{dx} = \frac{1}{2}(4) + 4 = 6$

3. Apply the chain rule to find the rate of change of depth

Using the chain rule, $\frac{dV}{dt} = \frac{dV}{dx} \times \frac{dx}{dt}$

$15 = 6 \times \frac{dx}{dt}$

$\frac{dx}{dt} = \frac{15}{6} = 2.5$