The figure below shows a pulley system where a conveyor belt is tied round the two wheels. The radius of the large wheel is 180cm and the distance between the centres of the wheels is 300cm and angle XOY = 140°. Determine (a) Length XV (b) Length VBW (c) The total length of the conveyor belt

Huu ni mfumo wa kapi wenye mikanda wazi. Kulingana na mchoro na taarifa, angle ya $140^\circ$ ni angle isiyogusana na mkanda kwenye kapi kubwa.

Data tuliyopewa:
Radius ya kapi kubwa, $R = 180 \text{ cm}$.
Umbali kati ya vituo vya kapi, $D = 300 \text{ cm}$.
Angle $\angle XOY = 140^\circ$.

Hatua ya 1: Tafuta angle $\alpha$ na radius ya kapi ndogo, $r$.
Angle ya kugusana na mkanda kwenye kapi kubwa ni $360^\circ - 140^\circ = 220^\circ$.
Kwa mkanda wazi, angle ya kugusana kwenye kapi kubwa ni $\theta_L = 180^\circ + 2\alpha$.
Hivyo,

$$220^\circ = 180^\circ + 2\alpha$$ $$2\alpha = 220^\circ - 180^\circ$$ $$2\alpha = 40^\circ$$ $$\alpha = 20^\circ$$

Sasa, tumia formula ya $\sin \alpha$ kwa mkanda wazi:

$$\sin \alpha = \frac{R-r}{D}$$ $$\sin 20^\circ = \frac{180 - r}{300}$$ $$180 - r = 300 \sin 20^\circ$$ $$r = 180 - 300 \sin 20^\circ$$ $$r = 180 - 300 \times 0.34202$$ $$r = 180 - 102.606$$ $$r = 77.394 \text{ cm}$$

a) Tafuta Urefu XV
Urefu XV ni urefu wa sehemu iliyonyooka ya mkanda.

$$XV = D \cos \alpha$$ $$XV = 300 \cos 20^\circ$$ $$XV = 300 \times 0.93969$$ $$XV = 281.907 \text{ cm}$$

Urefu XV $\approx \boxed{\text{281.91 cm}}$

b) Tafuta Urefu VBW
Urefu VBW ni urefu wa arc kwenye kapi ndogo.
Angle ya kugusana kwenye kapi ndogo ni $\theta_S = 180^\circ - 2\alpha$.

$$\theta_S = 180^\circ - 2(20^\circ) = 180^\circ - 40^\circ = 140^\circ$$

Badilisha angle kuwa radiani:

$$\theta_S = 140^\circ \times \frac{\pi}{180^\circ} = \frac{7\pi}{9} \text{ radiani}$$

Urefu VBW $= r \theta_S$

$$Urefu VBW = 77.394 \times \frac{7\pi}{9}$$ $$Urefu VBW \approx 77.394 \times 2.44346$$ $$Urefu VBW \approx 189.195 \text{ cm}$$

Urefu VBW $\approx \boxed{\text{189.20 cm}}$

c) Tafuta Urefu XAY
Urefu XAY ni urefu wa arc kwenye kapi kubwa.
Angle ya kugusana kwenye kapi kubwa ni $\theta_L = 220^\circ$.
Badilisha angle kuwa radiani:

$$\theta_L = 220^\circ \times \frac{\pi}{180^\circ} = \frac{11\pi}{9} \text{ radiani}$$

Urefu XAY $= R \theta_L$

$$Urefu XAY = 180 \times \frac{11\pi}{9}$$ $$Urefu XAY = 20 \times 11\pi = 220\pi \text{ cm}$$ $$Urefu XAY \approx 220 \times 3.14159$$ $$Urefu XAY \approx 691.150 \text{ cm}$$

Urefu XAY $\approx \boxed{\text{691.15 cm}}$

d) Tafuta Urefu wote wa mkanda wa kusafirisha
Urefu wote wa mkanda ni jumla ya sehemu mbili zilizonyooka na urefu wa arc kwenye kila kapi.

$$Urefu wote = 2 \times XV + \text{Urefu XAY} + \text{Urefu VBW}$$ $$Urefu wote = 2 \times 281.907 + 691.150 + 189.195$$ $$Urefu wote = 563.814 + 691.150 + 189.195$$ $$Urefu wote = 1444.159 \text{ cm}$$

Urefu wote $\approx \boxed{\text{1444.16 cm}}$