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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Huu hapa ni utatuzi wa kila safu katika jedwali: Safu ya 1: Radius kubwa (R) = 65 Radius ndogo (r) = 13 Urefu wa koni kubwa (H) = 25 Urefu wa koni ndogo (h) = 5 Step 1: Kokotoa urefu wa frustum. Urefu wa frustum (h_frustum) ni tofauti kati ya urefu wa koni kubwa na urefu wa koni ndogo. h_frustum = H - h = 25 - 5 = 20 Step 2: Tumia fomula ya ujazo wa frustum. Fomula ya ujazo wa frustum ni: V = (1)/(3) h_frustum (R^2 + Rr + r^2) Step 3: Badilisha thamani na kokotoa ujazo. V = (1)/(3) (20) (65^2 + 65 · 13 + 13^2) V = (20)/(3) (4225 + 845 + 169) V = (20)/(3) (5239) V = (104780)/(3) Safu ya 2: Radius kubwa (R) = 30 Urefu wa koni kubwa (H) = 60 Urefu wa koni ndogo (h) = 6 Step 1: Kokotoa radius ndogo (r). Tumia uwiano wa radius kwa urefu: (R)/(H) = (r)/(h). (30)/(60) = (r)/(6) r = (30 × 6)/(60) = (180)/(60) = 3 Radius ndogo iliyokosekana ni 3. Step 2: Kokotoa urefu wa frustum. h_frustum = H - h = 60 - 6 = 54 Step 3: Tumia fomula ya ujazo wa frustum. V = (1)/(3) h_frustum (R^2 + Rr + r^2) Step 4: Badilisha thamani na kokotoa ujazo. V = (1)/(3) (54) (30^2 + 30 · 3 + 3^2) V = 18 (900 + 90 + 9) V = 18 (999) V = 17982 Safu ya 3: Radius kubwa (R) = 140 Radius ndogo (r) = 21 Urefu wa koni kubwa (H) = 56 Step 1: Kokotoa urefu wa koni ndogo (h). Tumia uwiano wa radius kwa urefu: (R)/(H) = (r)/(h). (140)/(56) = (21)/(h) h = (21 × 56)/(140) = (1176)/(140) = 8.4 Urefu wa koni ndogo uliokosekana ni 8.4. Step 2: Kokotoa urefu wa frustum. h_frustum = H - h = 56 - 8.4 = 47.6 Step 3: Tumia fomula ya ujazo wa frustum. V = (1)/(3) h_frustum (R^2 + Rr + r^2) Step 4: Badilisha thamani na kokotoa ujazo. V = (1)/(3) (47.6) (140^2 + 140 · 21 + 21^2) V = (47.6)/(3) (19600 + 2940 + 441) V = (47.6)/(3) (22981) V = (1093307.6)/(3) Safu ya 4: Radius ndogo (r) = 18 Urefu wa koni kubwa (H) = 120 Urefu wa koni ndogo (h) = 40 Step 1: Kokotoa radius kubwa (R). Tumia uwiano wa radius kwa urefu: (R)/(H) = (r)/(h). (R)/(120) = (18)/(40) R = (18 × 120)/(40) = 18 × 3 = 54 Radius kubwa iliyokosekana ni 54. Step 2: Kokotoa urefu wa frustum. h_frustum = H - h = 120 - 40 = 80 Step 3: Tumia fomula ya ujazo wa frustum. V = (1)/(3) h_frustum (R^2 + Rr + r^2) Step 4: Badilisha thamani na kokotoa ujazo. V = (1)/(3) (80) (54^2 + 54 · 18 + 18^2) V = (80)/(3) (2916 + 972 + 324) V = (80)/(3) (4212) V = 80 (1404) V = 112320 That's 2 down. 3 left today — send the next one.