Tambua taarifa ulizopewa.

Mathematics

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2 parts
a

9.6 cm\text{9.6 cm}

b

20.93 cm2\text{20.93 cm}^2

Step-by-step solution

5 steps

Huu hapa ufumbuzi wa swali lako:

Part a· 3 steps

Kokotoa urefu wa kamba ya kawaida AB.

  1. Tambua taarifa ulizopewa.

    Tuna duara mbili zenye radius sawa, r=8r = 8 cm.
    Umbali kati ya vituo vya duara hizi ni d=12.8d = 12.8 cm.
    Kamba ya kawaida AB huunganisha sehemu mbili ambapo duara hizo hukutana.
    Mstari unaounganisha vituo vya duara (tuseme C1C_1 na C2C_2) hugawanya kamba ya kawaida AB katika sehemu mbili sawa na kwa pembe ya 90∘90^\circ.
    Kwa kuwa radius za duara zote ni sawa, sehemu ya katikati ya kamba AB (tuseme M) pia ni sehemu ya katikati ya mstari C1C2C_1C_2.
    Hivyo, umbali kutoka kituo kimoja hadi M ni h=d2=12.82=6.4h = \frac{d}{2} = \frac{12.8}{2} = 6.4 cm.

  2. Tumia theorem ya Pythagoras kupata nusu ya urefu wa kamba AB.

    Fikiria pembetatu ya kulia C1MAC_1MA, ambapo C1C_1 ni kituo cha duara, M ni sehemu ya katikati ya kamba AB, na A ni sehemu ya makutano.
    C1AC_1A ni radius, r=8r = 8 cm (hypotenuse).
    C1MC_1M ni umbali kutoka kituo hadi kamba, h=6.4h = 6.4 cm.
    AMAM ni nusu ya urefu wa kamba AB.
    Kutoka theorem ya Pythagoras:

    C1A2=C1M2+AM2C_1A^2 = C_1M^2 + AM^2 82=(6.4)2+AM28^2 = (6.4)^2 + AM^2 64=40.96+AM264 = 40.96 + AM^2 AM2=64−40.96AM^2 = 64 - 40.96 AM2=23.04AM^2 = 23.04 AM=23.04AM = \sqrt{23.04} AM=4.8 cmAM = 4.8 \text{ cm}
  3. Kokotoa urefu kamili wa kamba AB.

    Urefu wa kamba ya kawaida AB ni mara mbili ya AM.

    AB=2×AMAB = 2 \times AM AB=2×4.8 cmAB = 2 \times 4.8 \text{ cm} AB=9.6 cmAB = \boxed{\text{9.6 cm}}

Part b

20.93 cm2\text{20.93 cm}^2

Kokotoa eneo la kawaida kwa duara zote mbili.

  1. Kokotoa eneo la sehemu moja ya duara.

    Eneo la kawaida linaundwa na sehemu mbili zinazofanana za duara (circular segments), moja kutoka kila duara.
    Eneo la sehemu moja ya duara hutolewa na fomula:

    Asegment=r2cos⁡−1(hr)−hr2−h2A_{segment} = r^2 \cos^{-1}\left(\frac{h}{r}\right) - h\sqrt{r^2-h^2}

    Ambapo r=8r = 8 cm na h=6.4h = 6.4 cm.
    Kwanza, kokotoa angle θ=cos⁡−1(hr)\theta = \cos^{-1}\left(\frac{h}{r}\right) katika radiani:

    θ=cos⁡−1(6.48)=cos⁡−1(0.8)≈0.6435011 radiani\theta = \cos^{-1}\left(\frac{6.4}{8}\right) = \cos^{-1}(0.8) \approx 0.6435011 \text{ radiani}

    Kisha, kokotoa r2−h2\sqrt{r^2-h^2}:

    82−6.42=64−40.96=23.04=4.8 cm\sqrt{8^2 - 6.4^2} = \sqrt{64 - 40.96} = \sqrt{23.04} = 4.8 \text{ cm}

    Sasa badilisha thamani hizi kwenye fomula ya eneo la sehemu:

    Asegment=(82)×(0.6435011)−(6.4)×(4.8)A_{segment} = (8^2) \times (0.6435011) - (6.4) \times (4.8) Asegment=64×0.6435011−30.72A_{segment} = 64 \times 0.6435011 - 30.72 Asegment=41.1840704−30.72A_{segment} = 41.1840704 - 30.72 Asegment=10.4640704 cm2A_{segment} = 10.4640704 \text{ cm}^2
  2. Kokotoa eneo la kawaida.

    Eneo la kawaida ni mara mbili ya eneo la sehemu moja ya duara.

    Acommon=2×AsegmentA_{common} = 2 \times A_{segment} Acommon=2×10.4640704 cm2A_{common} = 2 \times 10.4640704 \text{ cm}^2 Acommon=20.9281408 cm2A_{common} = 20.9281408 \text{ cm}^2

    Kukadiria hadi sehemu mbili za desimali:

    Acommon≈20.93 cm2A_{common} \approx \boxed{\text{20.93 cm}^2}
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