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.

Mwelekeo wa duka la kwanza (R1) kutoka ghala (W) ni N75°W. Hii inamaanisha $75^\circ$ magharibi mwa kaskazini. Mwelekeo wa duka la pili (R2) kutoka ghala (W) ni $150^\circ$ (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 $360^\circ - 75^\circ = 285^\circ$.
• Pembe ya WR2 kutoka kaskazini (kisaa) ni $150^\circ$.
• Pembe $\angle R1WR2 = |285^\circ - 150^\circ| = 135^\circ$.

Tunayo pembetatu W-R1-R2. Umbali WR1 = 150 km. Umbali R1R2 = Kasi $\times$ Muda = $80 \text{ km/h} \times 3 \text{ h} = 240$ km. Pembe $\angle R1WR2 = 135^\circ$. Tunatafuta umbali WR2 (tuiite $d_2$).
Kwa kutumia Kanuni ya Kosine: $R1R2^2 = WR1^2 + WR2^2 - 2(WR1)(WR2) \cos(\angle R1WR2)$

Tumia fomula ya quadratic $d_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

Kwa kuwa umbali hauwezi kuwa hasi:

Umbali wa moja kwa moja kutoka duka la pili hadi ghala ni $\boxed{\text{109.2 km}}$.

• 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.

• 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.

• Umbali kutoka R2 hadi W = 109.22 km (kutoka sehemu a).
• Kasi ya kurudi = 60 km/h.
• Muda wa kusafiri kurudi W = Umbali / Kasi = $109.22 \text{ km} / 60 \text{ km/h} \approx 1.8203$ saa.
• Badilisha kuwa saa na dakika: $1.8203 \text{ h} = 1 \text{ h} + (0.8203 \times 60) \text{ min} \approx 1 \text{ h } 49 \text{ min}$.
• Lori liliondoka R2 saa 1:30 p.m.
• Muda wa kuwasili W = 1:30 p.m. + 1 saa 49 dakika = 3:19 p.m.

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 $\boxed{\text{3:19 p.m.}}$, ambayo ni mapema kuliko muda uliowekwa wa saa 5:30 p.m.