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.
ScanSolve AI Answer
a) (i) kimo AE, cha koni ndogo AXY. Step 1: Tambua uhusiano wa ulinganifu kati ya koni ndogo na koni kubwa. Koni AXY na koni ABC zinafanana. Hivyo, uwiano wa rediasi zao ni sawa na uwiano wa kimo chao. Tunajua rediasi ya koni ndogo r_1 = EX = 4 cm na rediasi ya koni kubwa r_2 = DB = 12 cm. Kimo cha koni ndogo ni AE = h_1. Kimo cha koni kubwa ni AD = h_2. Pia tunajua DE = h_2 - h_1 = 15 cm. (r_1)/(r_2) = (h_1)/(h_2) (4)/(12) = (h_1)/(h_2) (1)/(3) = (h_1)/(h_2) Hii inamaanisha h_2 = 3h_1. Step 2: Tafuta kimo cha koni ndogo (h_1). Tumia uhusiano h_2 - h_1 = 15 na badilisha h_2 na 3h_1. 3h_1 - h_1 = 15 2h_1 = 15 h_1 = (15)/(2) h_1 = 7.5 cm Kimo cha koni ndogo AE ni 7.5 cm. (ii) ujazo wa XBCY, umbo lililobaki. Step 3: Tafuta kimo cha koni kubwa (h_2). h_2 = 3h_1 = 3 × 7.5 h_2 = 22.5 cm Rediasi ya koni kubwa r_2 = 12 cm. Rediasi ya koni ndogo r_1 = 4 cm. Kimo cha koni ndogo h_1 = 7.5 cm. Step 4: Kokotoa ujazo wa koni kubwa ABC (V_2). Fomula ya ujazo wa koni ni V = (1)/(3) r^2 h. Tumia = 3.142. V_2 = (1)/(3) r_2^2 h_2 V_2 = (1)/(3) × 3.142 × (12)^2 × 22.5 V_2 = (1)/(3) × 3.142 × 144 × 22.5 V_2 = 3.142 × 48 × 22.5 V_2 = 3.142 × 1080 V_2 = 3393.36 cm^3 Step 5: Kokotoa ujazo wa koni ndogo AXY (V_1). V_1 = (1)/(3) r_1^2 h_1 V_1 = (1)/(3) × 3.142 × (4)^2 × 7.5 V_1 = (1)/(3) × 3.142 × 16 × 7.5 V_1 = 3.142 × 16 × 2.5 V_1 = 3.142 × 40 V_1 = 125.68 cm^3 Step 6: Kokotoa ujazo wa umbo lililobaki (frustum) XBCY. Ujazo wa frustum ni tofauti kati ya ujazo wa koni kubwa na ujazo wa koni ndogo. V_frustum = V_2 - V_1 V_frustum = 3393.36 - 125.68 V_frustum = 3267.68 cm^3 Ujazo wa XBCY ni 3267.68 cm^3. Tuma swali linalofuata 📸