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12.6E: Mazoezi ya Sehemu ya 12.6

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    178176

    • Edwin “Jed” Herman & Gilbert Strang
    • OpenStax
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    Kwa mazoezi 1 - 6, mchoro na kuelezea uso wa cylindrical wa equation iliyotolewa.

    1) [T]\( x^2+z^2=1\)

    Jibu

    Uso ni silinda na maamuzi yanayofanana na\(y\) -axis.

    2) [T]\( x^2+y^2=9\)

    3) [T]\( z=\cos\left(\frac{π}{2}+x\right)\)

    Jibu

    Uso ni silinda na maamuzi sambamba na\(y\) -axis.

    4) [T]\( z=e^x\)

    5) [T]\( z=9−y^2\)

    Jibu

    Uso ni silinda na maamuzi sambamba na\(x\) -axis.

    6) [T]\( z=\ln x\)

    Kwa mazoezi ya 7 - 10, grafu ya uso wa quadric hutolewa.

    Taja jina la uso wa quadric.

    b Kuamua mhimili wa ulinganifu wa uso wa quadric.

    7)

    Jibu
    a. silinda; b.\(x\) mhimili

    8)

    9)

    Jibu
    a. Hyperboloid ya karatasi mbili; b.\(x\) mhimili

    10)

    Kwa mazoezi 11 - 16, mechi ya uso wa quadric uliopewa na equation yake sambamba katika fomu ya kawaida.

    a.\( \dfrac{x^2}{4}+\dfrac{y^2}{9}−\dfrac{z^2}{12}=1\)

    b.\( \dfrac{x^2}{4}−\dfrac{y^2}{9}−\dfrac{z^2}{12}=1\)

    c.\( \dfrac{x^2}{4}+\dfrac{y^2}{9}+\dfrac{z^2}{12}=1\)

    d.\( z^2=4x^2+3y^2\)

    e.\( z=4x^2−y^2\)

    f.\( 4x^2+y^2−z^2=0\)

    11) Hyperboloid ya karatasi mbili

    Jibu
    b.

    12) Ellipsoid

    13) Paraboloid ya elliptic

    Jibu
    d.

    14) Paraboloid ya hyperbolic

    15) Hyperboloid ya karatasi moja

    Jibu
    a.

    16) koni ya elliptic

    Kwa mazoezi 17 - 28, fungua upya equation iliyotolewa ya uso wa quadric katika fomu ya kawaida. Tambua uso.

    17)\( −x^2+36y^2+36z^2=9\)

    Jibu
    \( −\dfrac{x^2}{9}+\dfrac{y^2}{\frac{1}{4}}+\dfrac{z^2}{\frac{1}{4}}=1,\)hyperboloid ya karatasi moja na\(x\) -axis kama mhimili wake wa ulinganifu

    18)\( −4x^2+25y^2+z^2=100\)

    19)\( −3x^2+5y^2−z^2=10\)

    Jibu
    \( −\dfrac{x^2}{\frac{10}{3}}+\dfrac{y^2}{2}−\dfrac{z^2}{10}=1,\)hyperboloid ya karatasi mbili na\(y\) -axis kama mhimili wake wa ulinganifu

    20)\( 3x^2−y^2−6z^2=18\)

    21)\( 5y=x^2−z^2\)

    Jibu
    \( y=−\dfrac{z^2}{5}+\dfrac{x^2}{5},\)paraboloid ya hyperbolic na\(y\) -axis kama mhimili wake wa ulinganifu

    22)\( 8x^2−5y^2−10z=0\)

    23)\( x^2+5y^2+3z^2−15=0\)

    Jibu
    \( \dfrac{x^2}{15}+\dfrac{y^2}{3}+\dfrac{z^2}{5}=1,\)ellipsoid

    24)\( 63x^2+7y^2+9z^2−63=0\)

    25)\( x^2+5y^2−8z^2=0\)

    Jibu
    \( \dfrac{x^2}{40}+\dfrac{y^2}{8}−\dfrac{z^2}{5}=0,\)koni ya elliptic na\(z\) -axis kama mhimili wake wa ulinganifu

    26)\( 5x^2−4y^2+20z^2=0\)

    27)\( 6x=3y^2+2z^2\)

    Jibu
    \( x=\dfrac{y^2}{2}+\dfrac{z^2}{3},\)elliptic paraboloid na\(x\) -axis kama mhimili wake wa ulinganifu

    28)\( 49y=x^2+7z^2\)

    Kwa mazoezi 29 - 34, tafuta maelezo ya uso uliotolewa wa quadric katika ndege maalum ya kuratibu na uifanye mchoro.

    29) [T]\( x^2+z^2+4y=0, \quad z=0\)

    Jibu

    Parabola\( y=−\frac{x^2}{4},\)

    30) [T]\( x^2+z^2+4y=0,\quad x=0\)

    31) [T]\( −4x^2+25y^2+z^2=100,\quad x=0\)

    Jibu

    duaradufu\( \dfrac{y^2}{4}+\dfrac{z^2}{100}=1,\)

    32) [T]\( −4x^2+25y^2+z^2=100,\quad y=0\)

    33) [T]\( x^2+\dfrac{y^2}{4}+\dfrac{z^2}{100}=1,\quad x=0\)

    Jibu

    duaradufu\( \dfrac{y^2}{4}+\dfrac{z^2}{100}=1,\)

    34) [T]\( x^2−y−z^2=1,\quad y=0\)

    35) Tumia grafu ya uso uliotolewa wa quadric ili kujibu maswali.

    Taja jina la uso wa quadric.

    b Ni ipi kati ya equations—\( 16x^2+9y^2+36z^2=3600, \; 9x^2+36y^2+16z^2=3600,\) au\( 36x^2+9y^2+16z^2=3600\) - sambamba na grafu?

    c Tumia b. kuandika equation ya uso quadric katika fomu ya kawaida.

    Jibu
    a. ellipsoid
    b. tatu equation
    c.\( \dfrac{x^2}{100}+\dfrac{y^2}{400}+\dfrac{z^2}{225}=1\)

    36) Tumia grafu ya uso uliotolewa wa quadric ili kujibu maswali.

    Taja jina la uso wa quadric.

    b Ni ipi kati ya equations—\( 36z=9x^2+y^2, \; 9x^2+4y^2=36z\), au\( −36z=−81x^2+4y^2\) - sambamba na grafu hapo juu?

    c Tumia b. kuandika equation ya uso quadric katika fomu ya kawaida.

    Kwa mazoezi 37 - 42, usawa wa uso wa quadric hutolewa.

    a Tumia njia ya kukamilisha mraba kuandika equation katika fomu ya kawaida.

    b Tambua uso.

    37)\( x^2+2z^2+6x−8z+1=0\)

    Jibu
    a.\(\dfrac{(x+3)^2}{16}+\dfrac{(z−2)^2}{8}=1\)
    b Silinda iliyozingatia\( (−3,2)\) na maamuzi sambamba na\(y\) -axis

    38)\( 4x^2−y^2+z^2−8x+2y+2z+3=0\)

    39)\( x^2+4y^2−4z^2−6x−16y−16z+5=0\)

    Jibu
    a.\(\dfrac{(x−3)^2}{4}+(y−2)^2−(z+2)^2=1\)
    b Hyperboloid ya karatasi moja iliyozingatia\( (3,2,−2),\) na\(z\) -axis kama mhimili wake wa ulinganifu

    40)\( x^2+z^2−4y+4=0\)

    41)\( x^2+\dfrac{y^2}{4}−\dfrac{z^2}{3}+6x+9=0\)

    Jibu
    a.\((x+3)^2+\dfrac{y^2}{4}−\dfrac{z^2}{3}=0\)
    b. koni ya elliptic iliyozingatia\( (−3,0,0),\) na\(z\) -axis kama mhimili wake wa ulinganifu

    42)\( x^2−y^2+z^2−12z+2x+37=0\)

    43) Andika fomu ya kawaida ya equation ya ellipsoid iliyozingatia asili ambayo hupita kupitia pointi\( A(2,0,0),\, B(0,0,1),\) na\( C\left(12,\sqrt{11},\frac{1}{2}\right).\)

    Jibu
    \( \dfrac{x^2}{4}+\dfrac{y^2}{16}+z^2=1\)

    44) Andika fomu ya kawaida ya equation ya ellipsoid katikati katika hatua\( P(1,1,0)\) ambayo hupita kupitia pointi\( A(6,1,0),\, B(4,2,0)\) na\( C(1,2,1)\).

    45) Kuamua pointi za makutano ya koni ya elliptic\( x^2−y^2−z^2=0\) na mstari wa equations ya ulinganifu\( \dfrac{x−1}{2}=\dfrac{y+1}{3}=z.\)

    Jibu
    \( (1,−1,0)\)na\( \left(\frac{13}{3},4,\frac{5}{3}\right)\)

    46) Kuamua pointi za makutano ya hyperboloid ya parabolic\( z=3x^2−2y^2\) na mstari wa equations parametric\( x=3t,\;y=2t,\;z=19t\), wapi\( t∈R.\)

    47) Pata equation ya uso wa quadric na pointi\( P(x,y,z)\) ambazo ni equidistant kutoka hatua\( Q(0,−1,0)\) na ndege ya equation\( y=1.\) Tambua uso.

    Jibu
    \( x^2+z^2+4y=0,\)elliptic paraboloid

    48) Pata equation ya uso wa quadric na pointi\( P(x,y,z)\) ambazo ni equidistant kutoka hatua\( Q(0,2,0)\) na ndege ya equation\( y=−2.\) Tambua uso.

    49) Kama uso wa kutafakari parabolic ni ilivyoelezwa na equation\( 400z=x^2+y^2,\) kupata hatua ya msingi ya reflector.

    Jibu
    \( (0,0,100)\)

    50) Fikiria reflector parabolic ilivyoelezwa na equation\( z=20x^2+20y^2.\) Pata hatua yake ya msingi.

    51) Onyesha kwamba uso wa quadric\( x^2+y^2+z^2+2xy+2xz+2yz+x+y+z=0\) hupunguza ndege mbili zinazofanana.

    52) Onyesha kwamba quadric uso\( x^2+y^2+z^2−2xy−2xz+2yz−1=0\) inapunguza ndege mbili sambamba kupita.

    53) [T] Makutano kati ya silinda\( (x−1)^2+y^2=1\) na nyanja\( x^2+y^2+z^2=4\) inaitwa Curve ya Viviani.

    a Tatua mfumo unao na equations ya nyuso ili kupata equation ya curve intersection. (Dokezo: Kupata\( x\) na\( y\) katika suala la\( z\).)

    b Tumia mfumo wa algebra ya kompyuta (CAS) au CalcPlot3D ili kutazama safu ya makutano kwenye nyanja\( x^2+y^2+z^2=4\).

    Jibu

    a.\(x=2−\dfrac{z^2}{2}, \quad y=±\dfrac{z}{2}\sqrt{4−z^2},\) wapi\( z∈[−2,2];\)

    b.

    54) Hyperboloid ya karatasi moja\( 25x^2+25y^2−z^2=25\) na koni ya elliptic\( −25x^2+75y^2+z^2=0\) inawakilishwa katika takwimu zifuatazo pamoja na curves zao za makutano. Tambua curves ya makutano na kupata equations yao (Dokezo: Tafuta y kutoka kwenye mfumo unao na usawa wa nyuso.)

    55) [T] Tumia CAS au CalcPlot3D kuunda makutano kati ya silinda\( 9x^2+4y^2=18\) na ellipsoid\( 36x^2+16y^2+9z^2=144\), na kupata equations ya curves makutano.

    Jibu

    ellipses mbili za equations\( \dfrac{x^2}{2}+\dfrac{y^2}{\frac{9}{2}}=1\) katika ndege\( z=±2\sqrt{2}\)

    56) [T] spheroid ni ellipsoid yenye semiaxes mbili sawa. Kwa mfano, equation ya spheroid na mhimili z kama mhimili wake wa ulinganifu hutolewa na\( \dfrac{x^2}{a^2}+\dfrac{y^2}{a^2}+\dfrac{z^2}{c^2}=1\), wapi\( a\) na\( c\) ni chanya namba halisi. Spheroid inaitwa oblate kama\( c<a\), na prolate kwa\( c>a\).

    a. kornea ya jicho inakadiriwa kama spheroid ya prolate yenye mhimili yaani jicho, wapi\( a=8.7mm\)\( c=9.6mm\) na.Andika equation ya spheroid ambayo inafanana na kamba na kuchora uso.

    b Kutoa mifano miwili ya vitu na maumbo ya spheroid ya prolate.

    57) [T] Katika masomo ya ramani, Dunia inakadiriwa na spheroid oblate badala ya nyanja. Radi kwenye ikweta na fito ni takriban\( 3963\) mi na\( 3950\) mi, mtawalia.

    Andika equation katika fomu ya kawaida ya ellipsoid ambayo inawakilisha sura ya Dunia. Fikiria katikati ya Dunia iko katika asili na kwamba mwelekeo uliotengenezwa\( z=0\) na ndege unafanana na ikweta.

    b. mchoro grafu.

    c Kupata equation ya Curve makutano ya uso na ndege\( z=1000\) ambayo ni sambamba na\(xy\) -ndege. Curve ya makutano inaitwa sambamba.

    d Pata usawa wa safu ya makutano ya uso na ndege\( x+y=0\) inayopita kupitia\(z\) -axis. Curve ya makutano inaitwa meridian.

    Jibu

    a.\(\dfrac{x^2}{3963^2}+\dfrac{y^2}{3963^2}+\dfrac{z^2}{3950^2}=1\)

    b.

    c. curve ya makutano ni duaradufu ya equation\( \dfrac{x^2}{3963^2}+\dfrac{y^2}{3963^2}=\dfrac{(2950)(4950)}{3950^2}\), na makutano ni duaradufu.
    d. curve makutano ni duaradufu ya equation\( \dfrac{2y^2}{3963^2}+\dfrac{z^2}{3950^2}=1.\)

    58) [T] seti ya buzzing kuhatarisha sumaku (au “rattlesnake mayai”) ni pamoja na mbili sparkling, polished, superstrong sumaku spheroid-umbo maalumu kwa ajili ya burudani ya watoto. Kila sumaku ni\( 1.625\) katika. muda mrefu na\( 0.5\) katika. pana katikati. Wakati wa kuwapiga ndani ya hewa, huunda sauti ya buzzing kama wanavyovutia.

    Andika equation ya spheroid prolate unaozingatia katika asili inayoelezea sura ya moja ya sumaku.

    Andika milinganyo ya spheroids ya prolate ambayo inafanana na sura ya sumaku za kuhatarisha za buzzing. Tumia CAS au CalcPlot3D ili kuunda grafu.

    59) [T] Uso wa umbo la moyo unatolewa na equation\( (x^2+\frac{9}{4}y^2+z^2−1)^3−x^2z^3−\frac{9}{80}y^2z^3=0.\)

    a Matumizi CAS au CalcPlot3D kwa graph uso kwamba mifano ya sura hii.

    b Kuamua na mchoro maelezo ya uso wa moyo juu ya\(xz\) -ndege.

    Jibu

    a.

    b. curve makutano ni\( (x^2+z^2−1)^3−x^2z^3=0.\)

    60) [T] pete torus ulinganifu kuhusu\(z\) -axis ni aina maalum ya uso katika topolojia na equation yake ni kutolewa na\( (x^2+y^2+z^2+R^2−r^2)^2=4R^2(x^2+y^2)\), ambapo\( R>r>0\). Nambari\( R\) na\( r\) zinaitwa ni radii kuu na ndogo, kwa mtiririko huo, ya uso. Takwimu inayofuata inaonyesha torus ya pete ambayo\( R=2\) na\( r=1\).

    Andika equation ya pete torus na\( R=2\) na\( r=1\), na kutumia CAS au CalcPlot3D kwa graph uso. Linganisha grafu na takwimu iliyotolewa.

    b Kuamua equation na mchoro maelezo ya torus pete kutoka a. juu ya\(xy\) -ndege.

    c Kutoa mifano miwili ya vitu na maumbo ya pete ya torus.