Math Problem Statement
Solution
We are given the equation of a parabola , and we need to find how fast the -coordinate is changing at the given point when the -coordinate is changing at a rate of 3 units per second.
Step 1: Implicit differentiation
Start by differentiating the given equation with respect to time . The equation is:
Differentiating both sides with respect to , using the chain rule:
Step 2: Solve for
Rearranging the equation to solve for :
Step 3: Substitute known values
The particle passes through the point , so and . We are also given that units per second.
Substitute these values into the equation:
Final Answer:
The rate at which the -coordinate is changing is units per second.
Would you like more details or have any questions?
Here are 5 questions related to this problem for further practice:
- How would the rate of change of vary if were different?
- What is the geometric interpretation of the relationship between and ?
- How does implicit differentiation apply to curves that are not functions?
- How can this method be applied to curves in 3D, such as surfaces?
- What happens if the point of interest on the curve is at the vertex of the parabola?
Tip: Always check that the point you are analyzing satisfies the given equation before performing differentiation.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Related Rates
Parabolas
Formulas
2y(dy/dt) = 4(dx/dt)
dx/dt = (y(dy/dt)) / 2
Theorems
Chain Rule of Differentiation
Suitable Grade Level
Grades 10-12
Related Recommendation
Related Rates Problem: Find Point on Curve y = x^2 - 1/4
Finding dx/dt for y = sqrt(x) at x = 1 with a given rate of dy/dt
Calculating Rate of Distance Change for a Point on Curve y = 4x
Related Rates: Distance Change for a Particle Moving Along y = sqrt(x)
Find the Vertical Rate of Change for a Particle on the Path x^3 - 3x^2 + 2