Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Related Rates

Calculus

4. Applications of Derivatives

Related Rates

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2:15 minutes

Problem 3.11.8

Textbook Question

At all times, the length of the long leg of a right triangle is 3 times the length x of the short leg of the triangle. If the area of the triangle changes with respect to time t, find equations relating the area A to x and dA/dt to dx/dt.

Verified step by step guidance

Start by expressing the relationship between the lengths of the legs of the triangle. Let the short leg be x, then the long leg is 3x.

The area A of a right triangle is given by the formula: A = (1/2) * base * height. Here, the base is x and the height is 3x, so A = (1/2) * x * 3x.

Simplify the expression for the area: A = (3/2) * x^2.

To find the relationship between dA/dt and dx/dt, use the chain rule for differentiation. Differentiate A with respect to t: dA/dt = d/dt[(3/2) * x^2].

Apply the chain rule: dA/dt = (3/2) * 2x * (dx/dt), which simplifies to dA/dt = 3x * (dx/dt). This equation relates the rate of change of the area to the rate of change of the short leg.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Area of a Right Triangle

The area A of a right triangle can be calculated using the formula A = (1/2) * base * height. In this case, the base can be considered the short leg (x) and the height the long leg, which is 3x. Therefore, the area can be expressed as A = (1/2) * x * (3x) = (3/2)x^2.

Related Rates

Related rates involve finding the relationship between the rates at which two or more quantities change. In this problem, we need to relate the rate of change of the area (dA/dt) to the rate of change of the short leg (dx/dt). This is typically done using implicit differentiation with respect to time.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. When applying the chain rule in this context, we differentiate the area A with respect to time t, leading to dA/dt = (dA/dx)(dx/dt). This allows us to express how the area changes as the length of the short leg changes.