Math Problem Statement
Solution
The problem asks how fast the distance of a particle from the origin is increasing. Let’s summarize and solve it step by step.
Problem summary:
- A particle moves in the -plane with the following rates of change:
- We need to determine the rate at which the distance from the origin is changing when the particle is at the point .
Solution:
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Distance from origin formula: where is the distance from the origin.
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Differentiate with respect to (using the chain rule):
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Substitute the values at the given point :
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Use the given rates and . Now substitute everything into the equation:
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Simplify the equation:
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Final answer:
The rate at which the particle's distance from the origin is increasing is approximately 1.46 m/s.
Would you like further clarifications, or do you have any additional questions?
Here are five follow-up questions related to this topic:
- How would the answer change if were positive?
- What does the sign of or tell us about the particle's movement?
- How can we find the angle of the particle's velocity with respect to the x-axis?
- What happens to if the particle were at the origin?
- How would the equation change if we were calculating speed instead of the rate of change of the distance?
Tip: Using the chain rule correctly is essential for related rates problems like these. Make sure to carefully identify what quantities are changing with time.
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Math Problem Analysis
Mathematical Concepts
Calculus
Related Rates
Distance Formula
Formulas
Distance formula: r = sqrt(x^2 + y^2)
Rate of change formula: dr/dt = (x/sqrt(x^2 + y^2)) * dx/dt + (y/sqrt(x^2 + y^2)) * dy/dt
Theorems
Chain Rule for Derivatives
Suitable Grade Level
Undergraduate or Advanced High School Calculus
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