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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4:09 minutes

Problem 3.8.32a

Textbook Question

Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.

a. How fast is the boat approaching the dock when 10 ft of rope are out?

Verified step by step guidance

Identify the right triangle formed by the rope, the vertical distance from the dock to the bow (6 ft), and the horizontal distance from the dock to the boat. Let x be the horizontal distance from the dock to the boat, and let L be the length of the rope.

Use the Pythagorean theorem to relate x, L, and the vertical distance: \( L^2 = x^2 + 6^2 \).

Differentiate both sides of the equation with respect to time t to find the relationship between the rates of change: \( 2L \frac{dL}{dt} = 2x \frac{dx}{dt} \).

Substitute the given values into the differentiated equation. You know \( \frac{dL}{dt} = -2 \) ft/sec (since the rope is being pulled in), L = 10 ft, and solve for \( \frac{dx}{dt} \), the rate at which the boat is approaching the dock.

Calculate x when L = 10 ft using the Pythagorean theorem: \( x = \sqrt{L^2 - 6^2} \). Substitute this value of x into the differentiated equation to find \( \frac{dx}{dt} \).

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Key Concepts

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

Related Rates

Related rates involve finding the rate at which one quantity changes with respect to another. In this problem, the rate at which the rope is hauled in affects the rate at which the dinghy approaches the dock. By using derivatives, we can relate these rates and solve for the unknown rate of the dinghy's movement.

Pythagorean Theorem

The Pythagorean Theorem is essential for relating the lengths in this problem. The rope forms the hypotenuse of a right triangle, with the vertical distance from the dock to the water and the horizontal distance from the dock to the dinghy as the other sides. This relationship helps in setting up the equation needed to find the rate at which the dinghy approaches the dock.

Differentiation

Differentiation is used to find the rate of change of a function. In this context, it helps determine how the length of the rope and the position of the dinghy change over time. By differentiating the equation derived from the Pythagorean Theorem, we can find the rate at which the dinghy approaches the dock when a specific length of rope is out.

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Textbook Question

Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.

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Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.a. How fast is the boat approaching the dock when 10 ft of rope are out?

Key Concepts

Related Rates

Pythagorean Theorem

Differentiation

Watch next

Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.

a. How fast is the boat approaching the dock when 10 ft of rope are out?