Contextual Applications of Differentiation

$100\pi \text{ cm}^3/\text{min}$

$30\text{ cm}^3/\text{min}$

$900\text{ cm}^3/\text{min}$

$300\text{ cm}^3/\text{min}$

The volume quadruples.

The volume remains the same.

The volume doubles.

The volume triples.

1:4

1:3

1:2

1:6

The shadow does not shrink.

$-0.8$m/s

$0$ m/s

$-1.6$m/s

The law of sines ($\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$).

The Pythagorean theorem ($a^2 + b^2 = c^2$).

The quadratic formula ($ax^2 + bx + c = 0$).

The distance formula ($\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$).

$V = \frac{1}{3} \pi r^2 h$

$V = \frac{4}{3} \pi r^2 h$

$V = \pi r^2 h$

$V = \pi r^2$

The radius is halved.

The radius is divided by $\sqrt{2}$.

The radius remains the same.

The radius is multiplied by $\sqrt{2}$.

$\frac{2}{\pi} \text{ cm/s}$

$\frac{1}{\pi} \text{ cm/s}$

$\frac{1}{2\pi} \text{ cm/s}$

$\frac{3}{4\pi} \text{ cm/s}$

Integration does not consider instant rates of change which are necessary to solve related rates problems.

Implicit differentiation directly relates all changing quantities through their derivatives.

Direct integration requires an initial condition that may not be provided or relevant to related rates problems.

Implicit differentiation applies when relationships between variables are given implicitly rather than explicitly.

Height

Radius

All of the above

Slant height