Lesson A sphere placed in an inverted cone
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<H2>A sphere placed in an inverted cone</H2> <H3>Problem 1</H3>A sphere is placed in an inverted hollow conical vessel of base radius 5 cm and vertical height 12 cm. If the highest point of the sphere is at the level of the base of the cone, find the radius of the sphere. <B>Solution</B> <pre> Let's consider vertical axial section of our configuration. In vertical section, we have inverted isosceles triangle ABC with the base BC at the top and vertex A. The base BC is 5 + 5 = 10 cm long. The height (or the altitude) of the triangle from A to BC is 12 cm. We also have a circle, inscribed into the triangle. This circle is the section of the sphere, placed inside the cone. Let start calculating the length of the lateral sides AB and AC. They are the hypotenuses of right angled triangles, so AB = AC = {{{sqrt(12^2 + 5^2)}}} = {{{sqrt(144+25)}}} = {{{sqrt(169)}}} = 13 cm. Obviously, the area of triangle ABC is {{{area[ABC]}}} = {{{(1/2)*10*12}}} = 60 square units. (1) (half the product the base and the altitude) We can calculate the area of triangle ABC in other way as half the product of its perimeter and the radius of this inscribed circle {{{area[ABC]}}} = {{{(1/2)*P*r}}}. (2) Left sides of equations (1) and (2) are equal ( both represent {{{area[ABC]}}} ). Therefore, we can write this equation 60 = {{{(1/2)*P*r}}}, or 120 = P*r (3) after reducing the factor 1/2 in both sides. The perimeter is easy to calculate: P = AB + AC + BC = 13 + 13 + 10 = 36 cm. So, equation (3) takes the form 120 = 36*r, which gives r = {{{120/36}}} = {{{10/3}}} = 3 {{{1/3}}} cm. Thus we found the radius of the circle and of the sphere. <U>ANSWER</U>. The radius of the sphere is 3 {{{1/3}}} cm. </pre> My other additional lessons on miscellaneous Geometry problems in this site are - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-rate-of-moving-of-the-tip-of-a-shadow.lesson>Find the rate of moving of the tip of a shadow</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-radio-transmitter-accessibility-area.lesson>A radio transmitter accessibility area</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-geometric-problems.lesson>Miscellaneous geometric problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-problems-on-parallelograms.lesson>Miscellaneous problems on parallelograms</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Remarkable-properties-of-triangles-into-which-diagonals-divide-a-quadrilateral.lesson>Remarkable properties of triangles into which diagonals divide a quadrilateral</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-trapezoid-divided-in-four-triangles-by-its-diagonals.lesson>A trapezoid divided in four triangles by its diagonals</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-problem-on-heptagon.lesson>A problem on a regular heptagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-area-of-a-regular-octagon.lesson>The area of a regular octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-fraction-of-the-area-of-a-regular-octagon.lesson>The fraction of the area of a regular octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Try-to-solve-this-nice-Geometry-problem.lesson>Try to solve these nice Geometry problems !</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-angle-between-sides-of-folded-triangle.lesson>Find the angle between sides of folded triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-problems-on-three-spheres.lesson>A problem on three spheres</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/07-An-upper-level-Geometry-problem-on-special-%2815-30-135%29-triangle.lesson>An upper level Geometry problem on special (15°,30°,135°)-triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-great-Math-Olympiad-level-Geometry-problem.lesson>A great Math Olympiad level Geometry problem</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Nice-geometry-problem-of-a-Math-Olympiad-level.lesson>Nice geometry problem of a Math Olympiad level</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/OVERVIEW-of-my-lessons-on-additional-misc-Geometry-problems.lesson>OVERVIEW of my additional lessons on miscellaneous advanced Geometry problems</A> To navigate over all topics/lessons of the Online Geometry Textbook use this file/link <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.
