Can anyone see if this is right? (problem with discs and parabolas)

  • Thread starter jaumzaum
  • Start date
In summary, the conversation discusses a challenge proposed by a teacher, which involves finding two points outside of two disks in the plane. The goal is to prove that all parabolas passing through these two points will intersect at least one of the disks. One person presents a potential solution involving choosing two points with the same slope on each disk, making it impossible for a tangent parabola to be drawn. However, another person points out that even a slight movement away from these points would allow for a parabola to be drawn that avoids both disks. Overall, the conversation focuses on the technicalities and intricacies of this problem.
  • #1
jaumzaum
434
33
My teacher proposed the following problem in the classroom as a challenge:
Consider 2 disks in the plane. Prove that you can always find 2 points exterior to the disks so that all the parabolas that pass through both points will intersect at least one of the disks.

I think I found a solution, can anyone check if it is right?
246465

246466
 
Physics news on Phys.org
  • #2
I'm not sure about this.

What about this reasoning:
No two points on a parabola will have the same slope.
Therefore, picking two points - each on one of the spheres and each with the same slope will make it impossible for a parabola to be drawn that is tangent at both points. Thus any parabola passing through those two points will cross into one of the circles.

But, having picked those two points, moving the slightest distance away will both bring you into compliance with the rules - and allow a parabola to be drawn that avoids both circles.

It seems like a very marginal situation. You create the possibility of avoiding both disks as soon as you become "exterior" to the disk.
 

1. Can you explain the problem with discs and parabolas?

The problem with discs and parabolas is a common topic in mathematics and physics. It involves finding the volume of a solid formed by rotating a parabola around a central axis. This problem is often used to illustrate the concept of integration and is important in many real-world applications.

2. How do I know if my solution is correct?

To determine if your solution is correct, you can check your work using mathematical formulas and equations. You can also use online calculators or ask a peer or teacher to review your solution. Additionally, you can compare your solution to known solutions or use graphing software to visualize the problem.

3. What are the key steps in solving this problem?

The key steps in solving the problem with discs and parabolas include identifying the parabola, setting up the integral, integrating to find the volume, and evaluating the integral. It is also important to pay attention to the limits of integration and any other given information, such as the radius of the disc or the axis of rotation.

4. Are there any common mistakes to watch out for?

Yes, there are a few common mistakes that can occur when solving this problem. One is forgetting to square the radius when setting up the integral. Another is using the wrong limits of integration, which can result in an incorrect answer. It is also important to double-check all calculations and make sure the units are consistent throughout the problem.

5. Can you provide an example of solving this problem?

Sure, here is an example of solving the problem with discs and parabolas: Find the volume of the solid formed by rotating the parabola y = x^2 from x = 0 to x = 2 around the x-axis.
Step 1: Identify the parabola and axis of rotation.
Step 2: Set up the integral: ∫πr^2dx, where r = y = x^2 and the limits of integration are 0 and 2.
Step 3: Integrate: ∫πx^4dx = πx^5/5 evaluated from 0 to 2.
Step 4: Evaluate the integral: π(2)^5/5 - π(0)^5/5 = 32π/5.
Therefore, the volume of the solid is 32π/5 cubic units.

Similar threads

Show uniqueness in Dirichlet problem in unit disk
    • Calculus and Beyond Homework Help
Replies
6
Views
366
Find the equation of a tangent line to y = x^2?
    • Calculus and Beyond Homework Help
Replies
11
Views
4K
Elliptic functions, diff eq, why proof on open disc holds for C
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Given a set of solids, compute the inward flux
    • Calculus and Beyond Homework Help
Replies
3
Views
933
Kinematic Problem w/ Parabola: Solving w/ KE Theorem?
    • Introductory Physics Homework Help
Replies
25
Views
2K
Can anyone please verify/check if there's error in this word problem?
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Simple parabola solving problem
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Electric Field due to a disk of radius R in the xy-plane
    • Advanced Physics Homework Help
Replies
7
Views
1K
Equations of two Lines that are Tangent to a Parabola
    • Calculus and Beyond Homework Help
Replies
8
Views
3K
Observation about the rotation of a disc
    • Mechanics
Replies
4
Views
721

Hot Threads

Recent Insights

Back
Top