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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Ili kukokotoa ujazo wa frustum, tutatoa ujazo wa koni ndogo iliyokatwa kutoka kwenye ujazo wa koni kubwa ya asili. Kutokana na maelezo: • Urefu wa frustum (H_f) = 40 cm • Radiasi kubwa (R) = 30 cm • Radiasi ndogo (r) = 7 cm • Urefu wa koni ndogo iliyokatwa (h) = 10 cm Hii inamaanisha: • Koni kubwa ina radiasi R = 30 cm na urefu jumla H = H_f + h = 40 cm + 10 cm = 50 cm. • Koni ndogo ina radiasi r = 7 cm na urefu h = 10 cm. Fomula ya ujazo wa koni ni V = (1)/(3) r^2 h. Tutatumia = (22)/(7). Step 1: Kokotoa ujazo wa koni kubwa. Ujazo wa koni kubwa (V_L) hutumia radiasi R = 30 cm na urefu H = 50 cm. V_L = (1)/(3) R^2 H V_L = (1)/(3) × (22)/(7) × (30 cm)^2 × (50 cm) V_L = (1)/(3) × (22)/(7) × 900 cm^2 × 50 cm V_L = (22)/(7) × 300 cm^2 × 50 cm V_L = (22)/(7) × 15000 cm^3 V_L = (330000)/(7) cm^3 V_L ≈ 47142.8571 cm^3 Step 2: Kokotoa ujazo wa koni ndogo iliyokatwa. Ujazo wa koni ndogo (V_S) hutumia radiasi r = 7 cm na urefu h = 10 cm. V_S = (1)/(3) r^2 h V_S = (1)/(3) × (22)/(7) × (7 cm)^2 × (10 cm) V_S = (1)/(3) × (22)/(7) × 49 cm^2 × 10 cm V_S = (1)/(3) × 22 × 7 cm^2 × 10 cm V_S = (1540)/(3) cm^3 V_S ≈ 513.3333 cm^3 Step 3: Kokotoa ujazo wa frustum. Ujazo wa frustum (V_f) ni tofauti kati ya ujazo wa koni kubwa na ujazo wa koni ndogo. V_f = V_L - V_S V_f = (330000)/(7) cm^3 - (1540)/(3) cm^3 Ili kutoa, tunahitaji kupata kigawo cha kawaida, ambacho ni 21. V_f = (330000 × 3)/(21) cm^3 - (1540 × 7)/(21) cm^3 V_f = (990000)/(21) cm^3 - (10780)/(21) cm^3 V_f = (990000 - 10780)/(21) cm^3 V_f = (979220)/(21) cm^3 V_f ≈ 46629.5238 cm^3 Ukizungusha hadi sehemu mbili za desimali: V_f ≈ 46629.52 cm^3 Ujazo wa frustum ni: 46629.52 cm^3 Tuma swali linalofuata 📸