Using appropriate mathematical reasoning: (a) Determine the distance of the direct route from the second retail outlet back to the warehouse. (b) Advise the transport manager whether the truck will return to the warehouse by 5:30 p.m. Justify your answer using calculations.

Question

Asked on July 16, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-07-16T19:45:29.921Z

Huu hapa ni utatuzi wa swali lako: a) Pata umbali wa moja kwa moja kutoka duka la pili hadi ghala. Step 1: Pata pembe kati ya njia za ghala-duka la kwanza na ghala-duka la pili. Mwelekeo wa duka la kwanza (R1) kutoka ghala (W) ni N75°W. Hii inamaanisha 75^ magharibi mwa kaskazini. Mwelekeo wa duka la pili (R2) kutoka ghala (W) ni 150^ (kutoka kaskazini kisaa). Pembe kati ya WR1 na WR2 kwenye ghala (W) ni tofauti ya pembe hizi kutoka kaskazini. Pembe ya WR1 kutoka kaskazini (kisaa) ni 360^ - 75^ = 285^. Pembe ya WR2 kutoka kaskazini (kisaa) ni 150^. Pembe R1WR2 = |285^ - 150^| = 135^. Step 2: Tumia Kanuni ya Kosine kupata umbali wa moja kwa moja kutoka R2 hadi W. Tunayo pembetatu W-R1-R2. Umbali WR1 = 150 km. Umbali R1R2 = Kasi × Muda = 80 km/h × 3 h = 240 km. Pembe R1WR2 = 135^. Tunatafuta umbali WR2 (tuiite d_2). Kwa kutumia Kanuni ya Kosine: R1R2^2 = WR1^2 + WR2^2 - 2(WR1)(WR2) ( R1WR2) 240^2 = 150^2 + d_2^2 - 2(150)(d_2) (135^) 57600 = 22500 + d_2^2 - 300 d_2 (-sqrt(2)2) 57600 = 22500 + d_2^2 + 150sqrt(2) d_2 d_2^2 + 150sqrt(2) d_2 - 35100 = 0 Tumia fomula ya quadratic d_2 = -b ± sqrt(b^2 - 4ac)2a: d_2 = -150sqrt(2) ± sqrt((1502))^2 - 4(1)(-35100)2 d_2 = -212.132 ± sqrt(45000 + 140400)2 d_2 = -212.132 ± sqrt(185400)2 d_2 = (-212.132 ± 430.581)/(2) Kwa kuwa umbali hauwezi kuwa hasi: d_2 = (-212.132 + 430.581)/(2) = (218.449)/(2) ≈ 109.22 km Umbali wa moja kwa moja kutoka duka la pili hadi ghala ni 109.2 km. b) Mshauri meneja wa usafiri kama lori litarejea ghala ifikapo saa 5:30 p.m. Step 3: Kokotoa muda wa kuondoka kwenye duka la kwanza (R1). Lori liliondoka ghala saa 7:00 a.m. Liliwasili R1 saa 8:45 a.m. Muda wa kupakua na kuangalia hisa R1 = dakika 45. Muda wa kuondoka R1 = 8:45 a.m. + 45 dakika = 9:30 a.m. Step 4: Kokotoa muda wa kuondoka kwenye duka la pili (R2). Lori liliondoka R1 saa 9:30 a.m. Muda wa kusafiri kutoka R1 hadi R2 = 3 saa. Muda wa kuwasili R2 = 9:30 a.m. + 3 saa = 12:30 p.m. Muda wa kupakua R2 = 1 saa. Muda wa kuondoka R2 = 12:30 p.m. + 1 saa = 1:30 p.m. Step 5: Kokotoa muda wa kusafiri kurudi ghala (W) na muda wa kuwasili. Umbali kutoka R2 hadi W = 109.22 km (kutoka sehemu a). Kasi ya kurudi = 60 km/h. Muda wa kusafiri kurudi W = Umbali / Kasi = 109.22 km / 60 km/h ≈ 1.8203 saa. Badilisha kuwa saa na dakika: 1.8203 h = 1 h + (0.8203 × 60) min ≈ 1 h 49 min. Lori liliondoka R2 saa 1:30 p.m. Muda wa kuwasili W = 1:30 p.m. + 1 saa 49 dakika = 3:19 p.m. Step 6: Linganisha muda wa kuwasili na muda unaohitajika. Muda wa kuwasili ghala ni 3:19 p.m. Muda unaohitajika wa kurudi ni kabla ya 5:30 p.m. Kwa kuwa 3:19 p.m. ni kabla ya 5:30 p.m., lori litarejea ghala kwa wakati. Lori litarejea ghala kabla ya saa 5:30 p.m. Lori litawasili ghala saa 3:19 p.m., ambayo ni mapema kuliko muda uliowekwa wa saa 5:30 p.m. Tuma swali linalofuata 📸

Accepted Answer · 2026-07-16T19:45:29.921Z

Huu hapa ni utatuzi wa swali lako: a) Pata umbali wa moja kwa moja kutoka duka la pili hadi ghala. Step 1: Pata pembe kati ya njia za ghala-duka la kwanza na ghala-duka la pili. Mwelekeo wa duka la kwanza (R1) kutoka ghala (W) ni N75°W. Hii inamaanisha 75^ magharibi mwa kaskazini. Mwelekeo wa duka la pili (R2) kutoka ghala (W) ni 150^ (kutoka kaskazini kisaa). Pembe kati ya WR1 na WR2 kwenye ghala (W) ni tofauti ya pembe hizi kutoka kaskazini. Pembe ya WR1 kutoka kaskazini (kisaa) ni 360^ - 75^ = 285^. Pembe ya WR2 kutoka kaskazini (kisaa) ni 150^. Pembe R1WR2 = |285^ - 150^| = 135^. Step 2: Tumia Kanuni ya Kosine kupata umbali wa moja kwa moja kutoka R2 hadi W. Tunayo pembetatu W-R1-R2. Umbali WR1 = 150 km. Umbali R1R2 = Kasi × Muda = 80 km/h × 3 h = 240 km. Pembe R1WR2 = 135^. Tunatafuta umbali WR2 (tuiite d_2). Kwa kutumia Kanuni ya Kosine: R1R2^2 = WR1^2 + WR2^2 - 2(WR1)(WR2) ( R1WR2) 240^2 = 150^2 + d_2^2 - 2(150)(d_2) (135^) 57600 = 22500 + d_2^2 - 300 d_2 (-sqrt(2)2) 57600 = 22500 + d_2^2 + 150sqrt(2) d_2 d_2^2 + 150sqrt(2) d_2 - 35100 = 0 Tumia fomula ya quadratic d_2 = -b ± sqrt(b^2 - 4ac)2a: d_2 = -150sqrt(2) ± sqrt((1502))^2 - 4(1)(-35100)2 d_2 = -212.132 ± sqrt(45000 + 140400)2 d_2 = -212.132 ± sqrt(185400)2 d_2 = (-212.132 ± 430.581)/(2) Kwa kuwa umbali hauwezi kuwa hasi: d_2 = (-212.132 + 430.581)/(2) = (218.449)/(2) ≈ 109.22 km Umbali wa moja kwa moja kutoka duka la pili hadi ghala ni 109.2 km. b) Mshauri meneja wa usafiri kama lori litarejea ghala ifikapo saa 5:30 p.m. Step 3: Kokotoa muda wa kuondoka kwenye duka la kwanza (R1). Lori liliondoka ghala saa 7:00 a.m. Liliwasili R1 saa 8:45 a.m. Muda wa kupakua na kuangalia hisa R1 = dakika 45. Muda wa kuondoka R1 = 8:45 a.m. + 45 dakika = 9:30 a.m. Step 4: Kokotoa muda wa kuondoka kwenye duka la pili (R2). Lori liliondoka R1 saa 9:30 a.m. Muda wa kusafiri kutoka R1 hadi R2 = 3 saa. Muda wa kuwasili R2 = 9:30 a.m. + 3 saa = 12:30 p.m. Muda wa kupakua R2 = 1 saa. Muda wa kuondoka R2 = 12:30 p.m. + 1 saa = 1:30 p.m. Step 5: Kokotoa muda wa kusafiri kurudi ghala (W) na muda wa kuwasili. Umbali kutoka R2 hadi W = 109.22 km (kutoka sehemu a). Kasi ya kurudi = 60 km/h. Muda wa kusafiri kurudi W = Umbali / Kasi = 109.22 km / 60 km/h ≈ 1.8203 saa. Badilisha kuwa saa na dakika: 1.8203 h = 1 h + (0.8203 × 60) min ≈ 1 h 49 min. Lori liliondoka R2 saa 1:30 p.m. Muda wa kuwasili W = 1:30 p.m. + 1 saa 49 dakika = 3:19 p.m. Step 6: Linganisha muda wa kuwasili na muda unaohitajika. Muda wa kuwasili ghala ni 3:19 p.m. Muda unaohitajika wa kurudi ni kabla ya 5:30 p.m. Kwa kuwa 3:19 p.m. ni kabla ya 5:30 p.m., lori litarejea ghala kwa wakati. Lori litarejea ghala kabla ya saa 5:30 p.m. Lori litawasili ghala saa 3:19 p.m., ambayo ni mapema kuliko muda uliowekwa wa saa 5:30 p.m. Tuma swali linalofuata 📸

Using appropriate mathematical reasoning: (a) Determine the distance of the direct route from the second retail outlet back to the warehouse. (b) Advise the transport manager whether the truck will return to the warehouse by 5:30 p.m. Justify your answer using calculations.

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Using appropriate mathematical reasoning: (a) Determine the distance of the direct route from the second retail outlet back to the warehouse. (b) Advise the transport manager whether the truck will return to the warehouse by 5:30 p.m. Justify your answer using calculations.

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