Geometrie-Stereometrie-Kegelstumpf


  • $Kegelstumpf$
    1 2 3 4

Beispiel Nr: 04
$ \text{Gegeben:}\\\text{Körperhöhe} \qquad h \qquad [m] \\ \text{Kreiszahl} \qquad \pi \qquad [] \\ \text{Radius 2} \qquad r_{2} \qquad [m] \\ \text{Radius 1} \qquad r_{1} \qquad [m] \\ \\ \text{Gesucht:} \\\text{Volumen} \qquad V \qquad [m^{3}] \\ \text{Oberfläche} \qquad O \qquad [m^{2}] \\ \\ Kegelstumpf\\ \textbf{Gegeben:} \\ h=6m \qquad \pi=3\frac{16}{113} \qquad r_{2}=20m \qquad r_{1}=40m \\ \\ \textbf{Rechnung:} \\ h=6m\\ \pi=3\frac{16}{113}\\ r_{2}=20m\\ r_{1}=40m\\ h_2=\dfrac{r_2\cdot h}{ r_1-r_2} \\ h_2=\dfrac{20m\cdot 6m}{ 40m-20m}=6m \\ h_1=h_2+h \\ h_1=6m+6m \\ \text{Pythagoras} \\ s_2=\sqrt{r_2^2+h_2^2} \quad s_1=\sqrt{r_1^2+h_1^2} \\ s_2=\sqrt{(20m)^2+(6m)^2}=20,9m \\ s_1=\sqrt{(40m)^2+(12m)^2}=41,8m \\ \text{Mantelfläche} \qquad M= r_1\cdot \pi \cdot s_1-r_2 \cdot \pi \cdot s_2 \\ M= 40m\cdot \pi \cdot 41,8m-20m \cdot \pi \cdot 20,9m=3,94\cdot 10^{3}m^{2} \\ \text{Grund- und Deckfläche} \qquad G= r_1^2\pi \quad D= r_2^2\pi \\ G= (40m)^2\pi=5,03\cdot 10^{3}m^{2} \\ D= (20m)^2\pi=1,26\cdot 10^{3}m^{2} \\ \text{Oberfläche} \qquad O= G+D+M \\ O= 5,03\cdot 10^{3}m^{2}+1,26\cdot 10^{3}m^{2}+3,94\cdot 10^{3}m^{2}=1,02\cdot 10^{4}m^{2} \\ \text{Volumen} \qquad V= \dfrac{1}{3} r_1^2\cdot \pi \cdot h_1 -\dfrac{1}{3} r_2^2\cdot \pi\cdot h_2 \\ V =\dfrac{1}{3} 40m ^{2} \cdot \pi \cdot 12m - \dfrac{1}{3} 20m ^{2} \cdot \pi\cdot 6m =1,76\cdot 10^{4}m^{3} \\\\\\ \small \begin{array}{|l|} \hline r1=\\ \hline 40 m \\ \hline 400 dm \\ \hline 4\cdot 10^{3} cm \\ \hline 4\cdot 10^{4} mm \\ \hline 4\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline r2=\\ \hline 20 m \\ \hline 200 dm \\ \hline 2\cdot 10^{3} cm \\ \hline 2\cdot 10^{4} mm \\ \hline 2\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h=\\ \hline 6 m \\ \hline 60 dm \\ \hline 600 cm \\ \hline 6\cdot 10^{3} mm \\ \hline 6\cdot 10^{6} \mu m \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline V=\\ \hline 1,76\cdot 10^{4} m^3 \\ \hline 17592919\frac{3}{25} dm^3 \\ \hline 1,76\cdot 10^{10} cm^3 \\ \hline 1,76\cdot 10^{13} mm^3 \\ \hline 17592919\frac{3}{25} l \\ \hline 1,76\cdot 10^{5} hl \\ \hline \end{array} \small \begin{array}{|l|} \hline s=\\ \hline 1,76\cdot 10^{4} m \\ \hline 1,76\cdot 10^{5} dm \\ \hline 1759291\frac{114}{125} cm \\ \hline 17592919\frac{3}{25} mm \\ \hline 1,76\cdot 10^{10} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h1=\\ \hline 12 m \\ \hline 120 dm \\ \hline 1,2\cdot 10^{3} cm \\ \hline 1,2\cdot 10^{4} mm \\ \hline 1,2\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h2=\\ \hline 6 m \\ \hline 60 dm \\ \hline 600 cm \\ \hline 6\cdot 10^{3} mm \\ \hline 6\cdot 10^{6} \mu m \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline s1=\\ \hline 41,8 m \\ \hline 418 dm \\ \hline 4,18\cdot 10^{3} cm \\ \hline 4,18\cdot 10^{4} mm \\ \hline 4,18\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline s2=\\ \hline 20,9 m \\ \hline 209 dm \\ \hline 2,09\cdot 10^{3} cm \\ \hline 2,09\cdot 10^{4} mm \\ \hline 2,09\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline M=\\ \hline 3,94\cdot 10^{3} m^2 \\ \hline 3,94\cdot 10^{5} dm^2 \\ \hline 3,94\cdot 10^{7} cm^2 \\ \hline 3,94\cdot 10^{9} mm^2 \\ \hline 39,4 a \\ \hline 0,394 ha \\ \hline \end{array} \small \begin{array}{|l|} \hline G=\\ \hline 5,03\cdot 10^{3} m^2 \\ \hline 502654\frac{104}{125} dm^2 \\ \hline 50265483\frac{1}{5} cm^2 \\ \hline 5,03\cdot 10^{9} mm^2 \\ \hline 50,3 a \\ \hline 0,503 ha \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline D=\\ \hline 1,26\cdot 10^{3} m^2 \\ \hline 1,26\cdot 10^{5} dm^2 \\ \hline 12566370\frac{4}{5} cm^2 \\ \hline 1,26\cdot 10^{9} mm^2 \\ \hline 12\frac{64}{113} a \\ \hline 0,126 ha \\ \hline \end{array} \small \begin{array}{|l|} \hline O=\\ \hline 1,02\cdot 10^{4} m^2 \\ \hline 1,02\cdot 10^{6} dm^2 \\ \hline 1,02\cdot 10^{8} cm^2 \\ \hline 1,02\cdot 10^{10} mm^2 \\ \hline 102 a \\ \hline 1,02 ha \\ \hline \end{array}$