1
vote
1answer
62 views

Maximum/Minimum of Curvature - Ellipse

Find the sum of the maximum and minimum of the curvature of the ellipse: $9(x-1)^2 + y^2 = 9$. Hint( Use the parametrization $x(t) = 1 + cos(t)$) Tried to use parametrization like that, but then ...
1
vote
2answers
119 views

Show that the intersection of a plane…

Show that the intersection of the plane $z = 2y$ with the elliptic cylinder $\frac{x^2}{5} + y^2 = 1$ is a circle. Find the radius and center of this circle. Hint: How can one describe a circle in ...
2
votes
2answers
49 views

Multivariable Calculus: Volume

Trying to figure out the following problem: Evaluate the integral $\int\int\int_EzdV$, where E lies above the paraboloid $z = x^2+y^2$ and below the plane $z=6y$. Round the result to the nearest ...
5
votes
1answer
571 views

How to maximize the volume of a rectangular parallelepiped in an ellipsoid?

This question comes from an exam about 15 years ago. How to find the maximal volume of a rectangular parallelepiped inscribed in an ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$? ...
0
votes
2answers
139 views

How many times can quadric kiss cosine at given point?

Let a quadric $ax^2+2bxy+cy^2+dx+ey+f=0$ touches the plot of $y=\cos(x)$ at the point $(0,1)$ with multiplicity $n$. What is the maximum possible value of $n$? Recall that a joint point $P$ of ...
1
vote
1answer
2k views

Parametric equation of a cone

I usually use the following parametric equation to find the surface area of a regular cone $z=\sqrt{x^2+y^2}$: $$x=r\cos\theta$$ $$y=r\sin\theta$$ $$z=r$$ And make $0\leq r \leq 2\pi$, $0 \leq \theta ...
2
votes
1answer
799 views

How to calculate minimum distance between two arbitrary ellipses in 2D?

Arbitrary ellipses means that they can be scaled, translated and rotated in any way in 2D. Do you know some high-school method (might be slightly more advanced than that) to find the minimum distance? ...