Geometrie-Stereometrie-Kegelstumpf

• $Kegelstumpf$
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Beispiel Nr: 02
$\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}=3m \qquad r_{1}=7m \\ \\ \textbf{Rechnung:} \\ h=6m\\ \pi=3\frac{16}{113}\\ r_{2}=3m\\ r_{1}=7m\\ h_2=\dfrac{r_2\cdot h}{ r_1-r_2} \\ h_2=\dfrac{3m\cdot 6m}{ 7m-3m}=4\frac{1}{2}m \\ h_1=h_2+h \\ h_1=4\frac{1}{2}m+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{(3m)^2+(4\frac{1}{2}m)^2}=5,41m \\ s_1=\sqrt{(7m)^2+(10\frac{1}{2}m)^2}=12,6m \\ \text{Mantelfläche} \qquad M= r_1\cdot \pi \cdot s_1-r_2 \cdot \pi \cdot s_2 \\ M= 7m\cdot \pi \cdot 12,6m-3m \cdot \pi \cdot 5,41m=227m^{2} \\ \text{Grund- und Deckfläche} \qquad G= r_1^2\pi \quad D= r_2^2\pi \\ G= (7m)^2\pi=154m^{2} \\ D= (3m)^2\pi=28,3m^{2} \\ \text{Oberfläche} \qquad O= G+D+M \\ O= 154m^{2}+28,3m^{2}+227m^{2}=409m^{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} 7m ^{2} \cdot \pi \cdot 10\frac{1}{2}m - \dfrac{1}{3} 3m ^{2} \cdot \pi\cdot 4\frac{1}{2}m =496m^{3} \\\\\\ \small \begin{array}{|l|} \hline r1=\\ \hline 7 m \\ \hline 70 dm \\ \hline 700 cm \\ \hline 7\cdot 10^{3} mm \\ \hline 7\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline r2=\\ \hline 3 m \\ \hline 30 dm \\ \hline 300 cm \\ \hline 3\cdot 10^{3} mm \\ \hline 3\cdot 10^{6} \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 496 m^3 \\ \hline 4,96\cdot 10^{5} dm^3 \\ \hline 496371646\frac{3}{5} cm^3 \\ \hline 4,96\cdot 10^{11} mm^3 \\ \hline 4,96\cdot 10^{5} l \\ \hline 4,96\cdot 10^{3} hl \\ \hline \end{array} \small \begin{array}{|l|} \hline s=\\ \hline 496 m \\ \hline 4,96\cdot 10^{3} dm \\ \hline 4,96\cdot 10^{4} cm \\ \hline 4,96\cdot 10^{5} mm \\ \hline 496371646\frac{3}{5} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h1=\\ \hline 10\frac{1}{2} m \\ \hline 105 dm \\ \hline 1,05\cdot 10^{3} cm \\ \hline 1,05\cdot 10^{4} mm \\ \hline 1,05\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h2=\\ \hline 4\frac{1}{2} m \\ \hline 45 dm \\ \hline 450 cm \\ \hline 4,5\cdot 10^{3} mm \\ \hline 4,5\cdot 10^{6} \mu m \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline s1=\\ \hline 12,6 m \\ \hline 126 dm \\ \hline 1,26\cdot 10^{3} cm \\ \hline 1,26\cdot 10^{4} mm \\ \hline 1,26\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline s2=\\ \hline 5,41 m \\ \hline 54,1 dm \\ \hline 541 cm \\ \hline 5,41\cdot 10^{3} mm \\ \hline 5,41\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline M=\\ \hline 227 m^2 \\ \hline 2,27\cdot 10^{4} dm^2 \\ \hline 2,27\cdot 10^{6} cm^2 \\ \hline 2,27\cdot 10^{8} mm^2 \\ \hline 2,27 a \\ \hline 0,0227 ha \\ \hline \end{array} \small \begin{array}{|l|} \hline G=\\ \hline 154 m^2 \\ \hline 1,54\cdot 10^{4} dm^2 \\ \hline 1,54\cdot 10^{6} cm^2 \\ \hline 153938042\frac{3}{10} mm^2 \\ \hline 1,54 a \\ \hline 0,0154 ha \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline D=\\ \hline 28,3 m^2 \\ \hline 2,83\cdot 10^{3} dm^2 \\ \hline 2,83\cdot 10^{5} cm^2 \\ \hline 28274334\frac{3}{10} mm^2 \\ \hline 0,283 a \\ \hline 0,00283 ha \\ \hline \end{array} \small \begin{array}{|l|} \hline O=\\ \hline 409 m^2 \\ \hline 4,09\cdot 10^{4} dm^2 \\ \hline 4,09\cdot 10^{6} cm^2 \\ \hline 4,09\cdot 10^{8} mm^2 \\ \hline 4,09 a \\ \hline 0,0409 ha \\ \hline \end{array}$