Related Rates Flashcard

The volume of a sphere is given by the formula: $$V = \frac{4}{3} \times \text{π} \times r^3$$ , where r is the radius.

To find the rate of change of the volume of a sphere, use the formula: $$\frac{dV}{dt} = 4\text{π}r^2 \frac{dr}{dt}$$ , where $$\frac{dr}{dt}$$ is the rate of change of the radius.

The area of a circle is given by the formula: $$A = \text{π} \times r^2$$ , where r is the radius.

To find the rate of change of the area of a circle, use the formula: $$\frac{dA}{dt} = 2\text{π}r \frac{dr}{dt}$$ , where $$\frac{dr}{dt}$$ is the rate of change of the radius.

The volume of a cone is given by the formula: $$V = \frac{1}{3} \times \text{π} \times r^2 \times h$$ , where r is the radius and h is the height.

To find the rate of change of the height of a cone, use the formula: $$\frac{dh}{dt} = -\frac{3V}{\text{π}r^2} \frac{dV}{dt}$$ , where $$\frac{dV}{dt}$$ is the rate of change of volume.

The volume of a balloon is given by the formula: $$V = \frac{4}{3} \times \text{π} \times r^3$$ .

To find the rate of change of the diameter, use the formula: $$\frac{dD}{dt} = 2 \frac{dr}{dt}$$ , where $$D = 2r$$ .

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): $$c^2 = a^2 + b^2$$ .

Using implicit differentiation on the Pythagorean theorem, you can find the relationship: $$\frac{dc}{dt} = \frac{a}{c} \frac{da}{dt} + \frac{b}{c} \frac{db}{dt}$$ .