Covering three different spheres with a cone
A 3D Cartesian coordinate frame is attached to a table such that its plane is coincident with the surface of the table. Three spheres are placed on the table. The first is of radius and centered at . The second is of radius and centered at . The third is of radius and centered at . You want to cover the spheres with a right circular cone whose axis is perpendicular to the surface of the table (i.e. parallel to the axis), and whose semi-vertical angle (the angle between its axis and its curved surface) is , such that the cone is tangent to each of the spheres. How high above the table is the apex of this conical cover ?
The answer is 40.475.
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1 solution
(0,0,0), (17,0,0) and (0,15,0) are the 3 points where the spheres are touching the cone's base. Since the semi-vertical angle is 30°, the cone's curved surface made 60° with its base and is bisected into 30° by each of the hypothenuse of the right triangles with the spheres' radii as their heights and √3 time the heights as the base lengths from the points mentioned above.
With that knowledge, we can find the point on the cone's base where the axis stands on. Knowing that the circumference is outside the right triangle and 10√3 units away from (0,0,0), 7√3 units away from (17,0,0) and 5√3 units away from (0,15,0), the cone's height's base must be both inside the right triangle and x units away from (0,0,0), x+3√3 units away from (17,0,0) and x+5√3 units away from (0,15,0).
Using cosine rules on two of the subtriangles we found above, we get x = 6.04822889 or x = (17√65762 - 2624)√3 / 497 for its positive solution.
(x + 5√3)² = x² + 15² - 2(x)(15)(m)
AND
(x + 3√3)² = x² + 17² - 2(x)(17)(n)
with
m² + n² = 1
(m & n are the cosines of the two angles comprising the original big triangle's 90°).
The cone's height is just √3 time the radius of the cone, while that would be equal to the sum of both distances inside and outside the original triangle.
Answer
= √3 × (cone's radius)
= √3 × (inside distance + outside distance)
= √3 × (6.04822889 + 10√3)
= 6.04822889√3 + 30
= 40.4758398 units (apex to base distance)