Let `k` be the length of any edge of a regular tetrahedron (a tetrahedron whose edges are equal in length is called a regular tetrahedron). Show that the angle between any edge and a face not containing the edge is`cos^(-1)(1//sqrt(3))` .

In a regular tetrahedron, prove that angle theta between any edge and the face not containing that edge is given by cos theta = 1/sqrt3 .

Prove that the smaller angle between any two diagonals of a cube is cos^(-1)""(1)/(3) .

The latest edge of a regular hexagonal pyramid is 1c mdot If the volume is maximum, then find its height.

The lengths of two opposite edges of a tetrahedron are a and b ; the shortest distane between these edges is d , and the angel between them is theta Prove using vectors that the volume of the tetrahedron is (a b dsi ntheta)/6 .

The volume of a tetrahedron fomed by the coterminus edges veca , vecb and vecc is 3 . Then the volume of the parallelepiped formed by the coterminus edges veca +vecb, vecb+vecc and vecc + veca is

Two concentric ellipses are such that the foci of one are on the other and their major axes are equal. Let ea n de ' be their eccentricities. Then. the quadrilateral formed by joining the foci of the two ellipses is a parallelogram the angle theta between their axes is given by theta=cos^(-1)sqrt(1/(e^2)+1/(e^('2))=1/(e^2e^('2))) If e^2+e^('2)=1, then the angle between the axes of the two ellipses is 90^0 none of these

Let P be any point on a directrix of an ellipse of eccentricity e ,S be the corresponding focus, and C the center of the ellipse. The line P C meets the ellipse at Adot The angle between P S and tangent a A is alpha . Then alpha is equal to (a) tan^(-1)e (b) pi/2 (c) tan^(-1)(1-e^2) (d) none of these

Let x be the length of one of the equal sides of an isosceles triangle, and let theta be the angle between them. If x is increasing at the rate (1/12) m/h, and theta is increasing at the rate of pi/(180) radius/h, then find the rate in m^3 / h at which the area of the triangle is increasing when x=12m and theta=pi//4.

Let x be the length of one of the equal sides of an isosceles triangle, and let theta be the angle between them. If x is increasing at the rate (1/12) m/h, and theta is increasing at the rate of pi/(180) radius/h, then find the rate in m^3 / h at which the area of the triangle is increasing when x=12ma n dtheta=pi//4.

O A B C is regular tetrahedron in which D is the circumcentre of O A B and E is the midpoint of edge A Cdot Prove that D E is equal to half the edge of tetrahedron.