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 $\approx \text{281.91 cm}$

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 $\approx \text{189.20 cm}$

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 $\approx \text{691.15 cm}$

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. $Urefu wote = 2 \times XV + Urefu XAY + \text{Urefu VBW}$ $Urefu wote = 2 \times 281.907 + 691.150 + 189.195$ $Urefu wote = 563.814 + 691.150 + 189.195$ $Urefu wote = 1444.159 \text{ cm}$ Urefu wote $\approx \text{1444.16 cm}$

That's 2 down. 3 left today — send the next one.