A circular mountain peak has coordinates (0,0,3) and its base has towns C=(-4,0,0), E=( 4,0,0) on opposite sides. Town D=(0,4,0) is halfway between them, around the base. A road goes partially up and down the mountain connecting C and E. Another road connects C and D. Both roads are shortest possible. What is the ratio of the road lengths?
An observer is standing on a plane at h metres below the centre of a reflecting sphere of radius r. Another object on the plane is perceived at an angle α from the vertical. What is α of the horizon in terms of h and r?
A tetrahedron with given vertex coordinates. What is its volume?
A pyramid with a square base and sides that are equilateral triangles contains a cube. If the cube is taking maximum space as shown, what is the fraction of its volume to the pyramid’s volume?
A cone has height 6 and base diameter 6. On it lie two points, one exactly halfway up, the other diametrically opposed at the base. What’s the shortest distance between these points going along the cone surface?
Two congruent pyramids placed at different distances from the observer appear as equilateral triangles in a square frame. If the closest one is at 100 meter, how far is the distant pyramid?
A spider situated at point A on the outside of a cylinder with diameter 4 and height 3 is trying to get to a fly at point C on the complete opposite side as fast as possible. What is the shortest route?
On two faces of a cube a diagonal is drawn that meet in the same vertex. What’s the angle between them?