U3P2 - Related Rates

The volume of a sphere is given by the formula: $$V = \frac{4}{3} \pi r^3$$ .

A related rates problem involves finding the rate at which one quantity changes with respect to another quantity that is also changing.

Use the chain rule: $$\frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}$$ .

The derivative is $$\frac{dV}{dr} = 4\pi r^2$$ .

Let $$\frac{dr}{dt} = 2 \text{ cm/sec}$$ .

The maximum height can be found by determining the vertex of the parabola represented by the position equation.

Use the formula $$t = -\frac{b}{2a}$$ from the quadratic equation $$s(t) = at^2 + bt + c$$ .

The coefficient of $$t^2$$ represents the acceleration due to gravity, while the coefficient of $$t$$ represents the initial velocity.

Set the position equation equal to zero and solve for $$t$$ .

In a right triangle formed by the ladder, wall, and ground, the Pythagorean theorem applies: $$x^2 + y^2 = L^2$$ , where $$L$$ is the length of the ladder.