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Keël

Keël

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Kegel
Oppervlak en VolumeGedeeltelike volumeAfleiding van formule vir volumeAfleiding van formule vir gedeeltelike volumeIn terme van R, H en hIn terme van R, H en rIn terme van R, r en hKyk ookEksterne skakels

'n Keël is 'n driedimensionele geometriese vorm wat deur twee parameters beskryf word. Dit kan vergelyk word met die vorm van 'n (afgeslote) heksehoed.

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'n Keël met radius r, hoogte h en lengte van skuinssy l.

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

Oppervlak kegel = π l r = π r ( r 2 + h 2 ) {\displaystyle {\text{Oppervlak kegel}}=\pi lr=\pi r{\sqrt {(r^{2}+h^{2})}}} {\displaystyle {\text{Oppervlak kegel}}=\pi lr=\pi r{\sqrt {(r^{2}+h^{2})}}}
Oppervlak basis = π r 2 {\displaystyle {\text{Oppervlak basis}}=\pi r^{2}} {\displaystyle {\text{Oppervlak basis}}=\pi r^{2}}
Totale oppervlak = π l r + π r 2 = π r ( r + ( r 2 + h 2 ) ) {\displaystyle {\text{Totale oppervlak}}=\pi lr+\pi r^{2}=\pi r\left(r+{\sqrt {(r^{2}+h^{2})}}\,\right)} {\displaystyle {\text{Totale oppervlak}}=\pi lr+\pi r^{2}=\pi r\left(r+{\sqrt {(r^{2}+h^{2})}}\,\right)}
Volume = 1 3 π r 2 h {\displaystyle {\text{Volume}}={\frac {1}{3}}\pi r^{2}h} {\displaystyle {\text{Volume}}={\frac {1}{3}}\pi r^{2}h}


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Keël.

Hier volg die oppervlak en volume van 'n keël.

Totale volume

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

Gedeeltelike volume

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

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

Gestel keël se dimensies is:

  • R = Radius
  • H = hoogte

Kies 'n baie dun skyfie in die keël wat afstand h van die bopunt is met dikte dh.

Die radius van die dun skyfie, kan soos volg bepaal word:

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

Die volume van die dun skyfie is die volgende:

d V = π r 2 d h = π ( R H h ) 2 d h = π ( R H ) 2 h 2 d h {\displaystyle dV=\pi r^{2}dh\quad =\quad \pi \left({\frac {R}{H}}h\right)^{2}dh\quad =\quad \pi \left({\frac {R}{H}}\right)^{2}h^{2}dh} {\displaystyle dV=\pi r^{2}dh\quad =\quad \pi \left({\frac {R}{H}}h\right)^{2}dh\quad =\quad \pi \left({\frac {R}{H}}\right)^{2}h^{2}dh}

Integreer nou van 0 tot H:

V = π ( R H ) 2 ∫ 0 H h 2 d h {\displaystyle V=\pi \left({\frac {R}{H}}\right)^{2}\int _{0}^{H}h^{2}dh} {\displaystyle V=\pi \left({\frac {R}{H}}\right)^{2}\int _{0}^{H}h^{2}dh}
V = π ( R H ) 2 1 3 [ H 2 − 0 2 ] {\displaystyle V=\pi \left({\frac {R}{H}}\right)^{2}{\frac {1}{3}}\left[H^{2}-0^{2}\right]} {\displaystyle V=\pi \left({\frac {R}{H}}\right)^{2}{\frac {1}{3}}\left[H^{2}-0^{2}\right]}
V = 1 3 π R 2 H {\displaystyle V={\frac {1}{3}}\pi R^{2}H} {\displaystyle V={\frac {1}{3}}\pi R^{2}H}

In terme van R, H en h

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

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

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

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

Hierdie kan weer terug vervang word in die oorspronklike formule:

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

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

Vervang weer terug in hoof formule:

Gedeeltelike volume = 1 3 π R 2 [ H − 1 H 2 ( H 3 − 3 H 2 h + 3 H h 2 − h 3 ) ] {\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 {\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]}

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

In terme van R, H en r

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

Stel:

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

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

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

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

Dus:

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

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


In terme van R, r en h

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

H R = H − h r {\displaystyle {\frac {H}{R}}={\frac {H-h}{r}}} {\displaystyle {\frac {H}{R}}={\frac {H-h}{r}}}
H R = H r − h r {\displaystyle {\frac {H}{R}}={\frac {H}{r}}-{\frac {h}{r}}} {\displaystyle {\frac {H}{R}}={\frac {H}{r}}-{\frac {h}{r}}}
H r − H R = h r {\displaystyle {\frac {H}{r}}-{\frac {H}{R}}={\frac {h}{r}}} {\displaystyle {\frac {H}{r}}-{\frac {H}{R}}={\frac {h}{r}}}
H ( 1 r − 1 R ) = h r {\displaystyle H\left({\frac {1}{r}}-{\frac {1}{R}}\right)={\frac {h}{r}}} {\displaystyle H\left({\frac {1}{r}}-{\frac {1}{R}}\right)={\frac {h}{r}}}
H ( R − r R r ) = h r {\displaystyle H\left({\frac {R-r}{Rr}}\right)={\frac {h}{r}}} {\displaystyle H\left({\frac {R-r}{Rr}}\right)={\frac {h}{r}}}
H = R r h r ( R − r ) {\displaystyle H={\frac {Rrh}{r\left(R-r\right)}}} {\displaystyle H={\frac {Rrh}{r\left(R-r\right)}}}
H = R h R − r {\displaystyle H={\frac {Rh}{R-r}}} {\displaystyle H={\frac {Rh}{R-r}}}

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

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

= 1 3 π R 3 h R − r ( 1 − r 3 R 3 ) {\displaystyle ={\frac {1}{3}}\pi R^{3}{\frac {h}{R-r}}\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.
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Oppervlak en Volume

Gedeeltelike volume

Afleiding van formule vir volume

Afleiding van formule vir gedeeltelike volume

Kyk ook

Eksterne skakels