Keël

Indien ${\displaystyle r}$ die straal van die sirkelvormige basis van die keël is, ${\displaystyle h}$ die hoogte is en ${\displaystyle l}$ die lengte van die skuinste van die punt van die keël tot die sirkelvormige rand, dan is:

${\displaystyle {\text{Oppervlak kegel}}=\pi lr=\pi r{\sqrt {(r^{2}+h^{2})}}}$

${\displaystyle {\text{Oppervlak basis}}=\pi r^{2}}$

${\displaystyle {\text{Totale oppervlak}}=\pi lr+\pi r^{2}=\pi r\left(r+{\sqrt {(r^{2}+h^{2})}}\,\right)}$

${\displaystyle {\text{Volume}}={\frac {1}{3}}\pi r^{2}h}$

Hier volg die oppervlak en volume van 'n keël.

Totale volume

${\displaystyle {\text{Totale volume}}={\frac {1}{3}}\pi R^{2}H}$

Gedeeltelike volume

• ${\displaystyle R}$ , ${\displaystyle H}$ : Radius en hoogte van groot keël
• ${\displaystyle r}$ : Klein radius
• ${\displaystyle h}$ : Afstand tussen groot en klein radius (sien skets)

In terme van R, H en h: ${\displaystyle {\text{Gedeeltelike volume}}=\pi {\frac {h}{H}}R^{2}\left(H-h+{\frac {h^{2}}{3H}}\right)}$

In terme van R, H en r: ${\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{2}H\left(1-{\frac {r^{3}}{R^{3}}}\right)}$

In terme van R, r en h: ${\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{3}{\frac {h}{R-r}}\left(1-{\frac {r^{3}}{R^{3}}}\right)}$

In terme van R, H en hWysig

${\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{2}H-{\frac {1}{3}}\pi r^{2}\left(H-h\right)}$

${\displaystyle ={\frac {1}{3}}\pi \left[R^{2}H-r^{2}\left(H-h\right)\right]}$

${\displaystyle r}$ kan soos volg geskryf word in terme van ${\displaystyle R}$ , ${\displaystyle H}$ en ${\displaystyle h}$ :

${\displaystyle {\frac {r}{H-h}}={\frac {R}{H}}}$

${\displaystyle r={\frac {R}{H}}\left(H-h\right)}$

Hierdie kan weer terug vervang word in die oorspronklike formule:

${\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi \left[R^{2}H-{\frac {R^{2}}{H^{2}}}\left(H-h\right)^{3}\right]}$

${\displaystyle \left(H-h\right)^{3}=H^{3}-3H^{2}h+3Hh^{2}-h^{3}}$

Vervang weer terug in hoof formule:

${\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{2}\left[H-{\frac {1}{H^{2}}}\left(H^{3}-3H^{2}h+3Hh^{2}-h^{3}\right)\right]}$

${\displaystyle ={\frac {1}{3}}\pi R^{2}\left[H-H+3h-3{\frac {h^{2}}{H}}+{\frac {h^{3}}{H^{2}}}\right]}$

${\displaystyle ={\frac {1}{3}}\pi hR^{2}\left[3-3{\frac {h}{H}}+{\frac {h^{2}}{H^{2}}}\right]}$

${\displaystyle =\pi hR^{2}\left[1-{\frac {h}{H}}+{\frac {h^{2}}{3H^{2}}}\right]}$

${\displaystyle =\pi {\frac {h}{H}}R^{2}\left[H-h+{\frac {h^{2}}{3H}}\right]}$

In terme van R, H en rWysig

${\displaystyle {\text{Gedeeltelike volume}}={\text{groot kegel}}-{\text{klein kegel}}}$

Stel:

• ${\displaystyle x}$ = hoogte van klein keël
• ${\displaystyle r}$ = radius van klein keël

${\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{2}H-{\frac {1}{3}}\pi r^{2}x}$

Die verhouding tussen ${\displaystyle x}$ , ${\displaystyle r}$ , ${\displaystyle H}$ en ${\displaystyle R}$ is:

${\displaystyle {\frac {x}{r}}={\frac {H}{R}}\qquad \Rightarrow \qquad x={\frac {rH}{R}}}$

Dus:

${\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{2}H-{\frac {1}{3}}\pi {\frac {r^{3}H}{R}}}$

${\displaystyle ={\frac {1}{3}}\pi R^{2}H\left(1-{\frac {r^{3}}{R^{3}}}\right)}$

In terme van R, r en hWysig

Kry eers H in terme van R, r en h:

${\displaystyle {\frac {H}{R}}={\frac {H-h}{r}}}$

${\displaystyle {\frac {H}{R}}={\frac {H}{r}}-{\frac {h}{r}}}$

${\displaystyle {\frac {H}{r}}-{\frac {H}{R}}={\frac {h}{r}}}$

${\displaystyle H\left({\frac {1}{r}}-{\frac {1}{R}}\right)={\frac {h}{r}}}$

${\displaystyle H\left({\frac {R-r}{Rr}}\right)={\frac {h}{r}}}$

${\displaystyle H={\frac {Rrh}{r\left(R-r\right)}}}$

${\displaystyle H={\frac {Rh}{R-r}}}$

Vervang hierdie nou in die formule wat in terme van R, H en r is:

${\displaystyle {\text{Gedeeltelike volume}}={\frac {1}{3}}\pi R^{2}H\left(1-{\frac {r^{3}}{R^{3}}}\right)}$

${\displaystyle ={\frac {1}{3}}\pi R^{3}{\frac {h}{R-r}}\left(1-{\frac {r^{3}}{R^{3}}}\right)}$

• Omtrek, oppervlakte en volume

• Wikiwoordeboek het 'n inskrywing vir keël.