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Sol cycled to raise funds for a charitable organization. On the first day, he cycled 48km for the first 10 days, he cycled 2km less on each subsequent day. Thereafter, he cycled 2km less on each subsequent day.

Mathematics

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3 parts
a(i)

22 km22 \text{ km}

a(ii)

10 km10 \text{ km}

b

Ksh 84000\text{Ksh 84000}

Step-by-step solution

7 steps

Huu hapa ni ufumbuzi wa swali lako:

Part a(i)· 2 steps

Umbali uliosafiriwa siku ya 10.

Kokotoa

  1. Tambua mfululizo wa hesabu.

    Kutokana na maelezo, Soi aliendesha baiskeli 40km siku ya kwanza. Kisha, aliendesha 2km pungufu kila siku inayofuata. Hii inaunda mfululizo wa hesabu (arithmetic progression) ambapo:
    Neno la kwanza (aa) = 40 km
    Tofauti ya kawaida (dd) = -2 km (kwa sababu umbali unapungua)
    Fomula ya neno la nn katika mfululizo wa hesabu ni an=a+(n−1)da_n = a + (n-1)d.

  2. Kokotoa umbali uliosafiriwa siku ya 10.

    Tumia n=10n=10 katika fomula:

    a10=40+(10−1)(−2)a_{10} = 40 + (10-1)(-2) a10=40+9(−2)a_{10} = 40 + 9(-2) a10=40−18a_{10} = 40 - 18 a10=22 kma_{10} = 22 \text{ km}

    Umbali uliosafiriwa siku ya 10 ni 22 km\boxed{22 \text{ km}}.

Part a(ii)

10 km10 \text{ km}

Umbali uliosafiriwa siku ya 16.

  1. Tumia fomula ya neno la nn katika mfululizo wa hesabu.

    Kama ilivyo hapo juu, a=40a = 40 km na d=−2d = -2 km.

  2. Kokotoa umbali uliosafiriwa siku ya 16.

    Tumia n=16n=16 katika fomula:

    a16=40+(16−1)(−2)a_{16} = 40 + (16-1)(-2) a16=40+15(−2)a_{16} = 40 + 15(-2) a16=40−30a_{16} = 40 - 30 a16=10 kma_{16} = 10 \text{ km}

    Umbali uliosafiriwa siku ya 16 ni 10 km\boxed{10 \text{ km}}.

Part b

Ksh 84000\text{Ksh 84000}

Ikiwa Soi alikusanya Ksh 200 kwa kila Km, kokotoa jumla ya pesa zilizokusanywa.

  1. Tambua idadi kamili ya siku za kuendesha baiskeli.

    Soi ataendelea kuendesha baiskeli hadi umbali uwe 0 au chini. Tutapata nn wakati an=0a_n = 0.

    an=40+(n−1)(−2)a_n = 40 + (n-1)(-2) 0=40−2n+20 = 40 - 2n + 2 0=42−2n0 = 42 - 2n 2n=422n = 42 n=21n = 21

    Kwa hivyo, Soi aliendesha baiskeli kwa siku 21.

  2. Kokotoa jumla ya umbali uliosafiriwa kwa siku zote.

    Jumla ya mfululizo wa hesabu ni Sn=n2(a+an)S_n = \frac{n}{2}(a + a_n).
    Hapa, n=21n=21, a=40a=40, na a21=0a_{21}=0 (umbali siku ya 21).

    S21=212(40+0)S_{21} = \frac{21}{2}(40 + 0) S21=212(40)S_{21} = \frac{21}{2}(40) S21=21×20S_{21} = 21 \times 20 S21=420 kmS_{21} = 420 \text{ km}
  3. Kokotoa jumla ya pesa zilizokusanywa.

    Kiwango cha kukusanya pesa ni Ksh 200 kwa kila Km.

    Jumla ya pesa=Jumla ya umbali×Kiwango\text{Jumla ya pesa} = \text{Jumla ya umbali} \times \text{Kiwango} Jumla ya pesa=420 km×Ksh 200/km\text{Jumla ya pesa} = 420 \text{ km} \times \text{Ksh 200/km} Jumla ya pesa=Ksh 84000\text{Jumla ya pesa} = \text{Ksh 84000}

    Jumla ya pesa zilizokusanywa ni Ksh 84000\boxed{\text{Ksh 84000}}.

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