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.
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Sawa blessed, hebu tutatue swali hili la jiometri. a) Kokotoa ujazo wa maji kwenye ndoo (frustum). Ndoo ni frustum yenye: Kipenyo cha juu (R) = 8 cm Kipenyo cha chini (r) = 6 cm Urefu wa frustum (h_f) = 30 cm Ili kukokotoa ujazo wa frustum, tunaweza kutumia fomula ya ujazo wa koni kubwa kutoa ujazo wa koni ndogo iliyokatwa. Kwanza, tunahitaji kupata urefu wa koni kubwa (H_L) na koni ndogo (H_S). Kwa kutumia pembetatu zinazofanana: (R)/(H_L) = (r)/(H_S) (8)/(H_L) = (6)/(H_S) 8H_S = 6H_L H_S = (6)/(8)H_L = (3)/(4)H_L Tunajua kuwa urefu wa frustum ni H_L - H_S = 30 cm. Badilisha H_S: H_L - (3)/(4)H_L = 30 (1)/(4)H_L = 30 H_L = 120 cm Sasa tafuta H_S: H_S = (3)/(4) × 120 = 90 cm Step 1: Kokotoa ujazo wa koni kubwa. V_L = (1)/(3) R^2 H_L V_L = (1)/(3) × (22)/(7) × (8 cm)^2 × (120 cm) V_L = (1)/(3) × (22)/(7) × 64 × 120 V_L = (22 × 64 × 40)/(7) = (56320)/(7) cm^3 Step 2: Kokotoa ujazo wa koni ndogo. V_S = (1)/(3) r^2 H_S V_S = (1)/(3) × (22)/(7) × (6 cm)^2 × (90 cm) V_S = (1)/(3) × (22)/(7) × 36 × 90 V_S = (22 × 12 × 90)/(7) = (23760)/(7) cm^3 Step 3: Kokotoa ujazo wa maji (ujazo wa frustum). V_w = V_L - V_S V_w = (56320)/(7) - (23760)/(7) V_w = (56320 - 23760)/(7) = (32560)/(7) cm^3 V_w = (32560)/(7) cm^3 ≈ 4651.43 cm^3 b) Maji yote yamemiminwa kwenye chombo cha cylindrical chenye kipenyo cha duara 8 cm na urefu wa 35 cm. Kokotoa urefu wa silinda juu ya maji. Chombo cha cylindrical kina: Kipenyo (r_c) = 8 cm Urefu (H_c) = 35 cm Step 1: Kokotoa urefu wa maji (h_w) kwenye silinda. Ujazo wa maji (V_w) ni sawa na ujazo wa frustum uliokokotolewa hapo juu. V_w = r_c^2 h_w (32560)/(7) = (22)/(7) × (8 cm)^2 × h_w (32560)/(7) = (22)/(7) × 64 × h_w Kuzidisha pande zote mbili kwa 7: 32560 = 22 × 64 × h_w 32560 = 1408 × h_w h_w = (32560)/(1408) h_w = 23.125 cm Step 2: Kokotoa urefu wa silinda juu ya maji. Urefu juu ya maji = Urefu wa silinda - Urefu wa maji H_above = H_c - h_w H_above = 35 cm - 23.125 cm H_above = 11.875 cm H_above = 11.875 cm 3 done, 2 left today. You're making progress.