๐ Understanding Related Rates
Related rates problems in calculus involve finding the rate at which one quantity is changing by relating it to other quantities whose rates of change are known. These problems often involve geometric shapes and their properties, such as area, volume, and perimeter. The key is to identify the relationships between the variables and use implicit differentiation to find the desired rate.
๐ Historical Context
The development of calculus, primarily by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, provided the tools necessary to analyze rates of change and solve related rates problems. Early applications were in physics and astronomy, but the techniques quickly spread to other fields, including engineering and economics.
๐ Key Principles
- ๐ Identify Variables and Rates: Determine which quantities are changing and what rates are given or need to be found.
- ๐ Establish a Relationship: Find an equation that relates the variables. This often involves geometric formulas (e.g., area of a circle, volume of a sphere).
- ๐ก Implicit Differentiation: Differentiate both sides of the equation with respect to time ($t$). Remember to apply the chain rule.
- ๐ข Substitute Known Values: Plug in the given rates and values at the specific instant in time.
- โ
Solve for the Unknown Rate: Solve the resulting equation for the rate you want to find.
๐ Area Applications
Let's explore some examples involving area:
- ๐ต Expanding Circle: Suppose a circle's radius is increasing at a rate of 3 cm/s. How fast is the area increasing when the radius is 10 cm?
- ๐ Area of a circle: $A = \pi r^2$
- ๐ก Differentiate with respect to time: $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$
- โ
Substitute given values: $\frac{dA}{dt} = 2\pi (10)(3) = 60\pi$ cm$^2$/s
- ๐ถ Changing Rectangle: A rectangle has a length that is increasing at 2 m/s and a width that is decreasing at 1 m/s. At what rate is the area changing when the length is 12 m and the width is 5 m?
- ๐ Area of a rectangle: $A = lw$
- ๐ก Differentiate with respect to time: $\frac{dA}{dt} = l\frac{dw}{dt} + w\frac{dl}{dt}$
- โ
Substitute given values: $\frac{dA}{dt} = (12)(-1) + (5)(2) = -12 + 10 = -2$ m$^2$/s
๐ฆ Volume Applications
Now, let's look at examples involving volume:
- ๐ Inflating Balloon: Air is being pumped into a spherical balloon at a rate of 100 cm$^3$/s. How fast is the radius increasing when the diameter is 50 cm?
- ๐ Volume of a sphere: $V = \frac{4}{3}\pi r^3$
- ๐ก Differentiate with respect to time: $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$
- โ
Substitute given values: $100 = 4\pi (25)^2 \frac{dr}{dt} \Rightarrow \frac{dr}{dt} = \frac{100}{4\pi (625)} = \frac{1}{25\pi}$ cm/s
- ๐ง Melting Ice Cube: A cube of ice is melting such that its volume is decreasing at a rate of 5 cm$^3$/min. How fast is the edge length decreasing when the edge length is 10 cm?
- ๐ Volume of a cube: $V = s^3$
- ๐ก Differentiate with respect to time: $\frac{dV}{dt} = 3s^2 \frac{ds}{dt}$
- โ
Substitute given values: $-5 = 3(10)^2 \frac{ds}{dt} \Rightarrow \frac{ds}{dt} = \frac{-5}{300} = -\frac{1}{60}$ cm/min
๐ Perimeter Applications
Finally, consider problems involving perimeter:
- ๐ Expanding Square: The side of a square is increasing at a rate of 4 in/min. How fast is the perimeter increasing when the side is 15 inches?
- ๐ Perimeter of a square: $P = 4s$
- ๐ก Differentiate with respect to time: $\frac{dP}{dt} = 4 \frac{ds}{dt}$
- โ
Substitute given values: $\frac{dP}{dt} = 4(4) = 16$ in/min
- ๐บ Equilateral Triangle: The side length of an equilateral triangle is decreasing at a rate of 2 cm/s. How fast is the perimeter decreasing when the side length is 8 cm?
- ๐ Perimeter of an equilateral triangle: $P = 3s$
- ๐ก Differentiate with respect to time: $\frac{dP}{dt} = 3 \frac{ds}{dt}$
- โ
Substitute given values: $\frac{dP}{dt} = 3(-2) = -6$ cm/s
๐ Practice Quiz
- โ A right triangle has legs of length $a$ and $b$. If $a$ is increasing at 3 m/s and $b$ is increasing at 4 m/s, how fast is the hypotenuse $c$ increasing when $a = 5$ m and $b = 12$ m?
- โ A conical tank (vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate at which the depth of the water is increasing when the water is 8 feet deep.
๐ก Conclusion
Related rates problems can seem daunting, but by breaking them down into smaller steps and understanding the underlying principles, you can tackle them with confidence. Remember to identify the variables, establish relationships, use implicit differentiation, and substitute known values to solve for the unknown rate. Keep practicing, and you'll master these problems in no time! ๐