Lesson A cylindrical glass filled with water and then tilted to a horizon
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<H2>A cylindrical glass filled with water and then tilted to a horizon</H2> <H3>Problem 1</H3>A cylindrical glass of (inside) diameter 6 cm and height 11 cm is filled and then tilted to a 45° angle, so that some water overflows. How much water is left in it? <B>Solution</B> <pre> <Table> <TR> <TD> In Figure, rectangle ABCD represents the tilted glass (or, more precisely, its axial section). Blue line ED represents the horizontal surface of liquid in the glass. Triangle AED is a right-angled triangle with angles AED and ADE of 45°, so triangle AED is the isosceles right-angled triangle. Thus, AE = AD = 6 cm. Red line EF in Figure is the trace of a plane parallel to the glass edge. This plane defines a cylinder AEFD. The surface (the plane) ED cuts cylinder AEFD in two parts, AED and EFD. Due to symmetry, the parts AED and EFD have equal volumes, each of which is half the volume of the cylinder AEFD. Cylinder AEFD has the radius of 3 cm and the height AE = AD = 6 cm, so the volume of cylinder AEFD is {{{V[AEFD]}}} = {{{pi*3^2*6}}} = {{{54pi}}}. </TD> <TD> {{{drawing( 400, 400, -2, 14, -6, 10, line( 0, 0, 7.778, 7.778), line( 7.778, 7.778, 12.02, 3.535), line( 0, 0, 4.242, -4.242), line( 4.242, -4.242, 12.02, 3.535), locate( 7.4, 8.4, A), locate(-0.6, 0.3, B), locate( 3.9, -4.3, C), locate( 12.2, 4, D), blue(line( 3.535, 3.535, 12.02, 3.535)), locate(3.0, 4, E), locate(7, 2.8, water), red(line( 3.535, 3.535, 7.777, -0.707)), locate(8.0, -0.65, F))}}} <B>Figure</B>. The tilted glass with water <B>ABCD</B> </TD> </TR> </TABLE> Therefore, the volume of liquid poured from the glass is {{{(54/2)pi}}} = {{{27pi}}} = {{{27*3.14159}}} = 84.82293 cm^3. The volume of the water remaining in the glass is {{{pi*r^2*h}}} - 84.82293 = {{{3.14159*3^2*11}}} - 84.82293 = 311.01741 - 84.82293 = 226.1945, or about 226.2 cm^3 (or 226.2 mL). <U>ANSWER</U>. The volume of water remaining in the glass is about 226.2 cm^3 (or 2226.2 mL). </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-square-divided-into-three-equal-areas-by-two-parallel-lines.lesson>A square divided into three equal areas by two parallel lines</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/A-problem-on-equiangular-but-not-equilateral-octagon.lesson>A problem on equiangular but not equilateral 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/A-sphere-placed-in-an-inverted-cone.lesson>A sphere placed in an inverted cone</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-partially-filled-cylindrical-tank-changes-position-from-horizontal-to-vertical.lesson>A partially filled cylindrical tank changes position from horizontal to vertical</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>.
