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Related Rates
How Quantities Change Together
Related Tools
Related Labs
Related rates is a calculus topic about quantities that change with time and are linked by an equation. Instead of finding how one variable changes by itself, you use a relationship between variables to connect their rates of change. This matters because many real situations involve several changing measurements at once, such as ladders sliding, balloons expanding, or shadows moving. The goal is usually to find one rate like dx/dt or dy/dt when another rate is known.
The main method is to write an equation that relates the variables, then differentiate both sides with respect to time t. This uses implicit differentiation because the variables depend on time even if t does not appear directly in the original equation. After differentiating, substitute the known values from the specific instant in the problem and solve for the unknown rate. Units and signs are important because they tell whether a quantity is increasing or decreasing.
Key Facts
- Related rates studies variables x, y, r, V, and others that all change with time t.
- Start with a geometric or physical relationship, such as x^2 + y^2 = L^2 for a ladder of fixed length.
- Differentiate with respect to time: d/dt(x^2 + y^2) = d/dt(L^2) gives 2x dx/dt + 2y dy/dt = 0 when L is constant.
- For a circle with changing radius, A = pi r^2 leads to dA/dt = 2pi r dr/dt.
- For a sphere with changing radius, V = (4/3)pi r^3 leads to dV/dt = 4pi r^2 dr/dt.
- Always evaluate rates at one instant by substituting both the current dimensions and the known rate values.
Vocabulary
- Related rates
- A calculus method for finding how one changing quantity depends on the rate of change of another connected quantity.
- Implicit differentiation
- A differentiation technique used when variables are related by an equation and each variable may depend on time.
- Rate of change
- The amount a quantity changes per unit time, often written as a derivative like dx/dt.
- Instant
- The specific moment at which the given measurements and rates are used in a related rates problem.
- Constant
- A quantity that does not change with time, so its derivative with respect to time is zero.
Common Mistakes to Avoid
- Using the given dimensions before differentiating, which is wrong because you must first keep variables general and differentiate the full relationship.
- Forgetting that every changing variable depends on time, which is wrong because terms like x and y need the chain rule and become dx/dt and dy/dt after differentiation.
- Ignoring signs on rates, which is wrong because decreasing quantities should have negative derivatives and the sign affects the final answer.
- Substituting an incorrect geometric relationship, which is wrong because the whole solution depends on starting from the correct equation such as x^2 + y^2 = L^2 for a right triangle.
Practice Questions
- 1 A 10 ft ladder leans against a wall. The bottom slides away from the wall at 2 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall?
- 2 The radius of a balloon increases at 0.5 cm/s. How fast is the volume changing when the radius is 4 cm? Use V = (4/3)pi r^3.
- 3 In a related rates problem, why must you substitute the numerical values only after differentiating the relationship between the variables?