The volume,V of a cylinder varies jointly as its height (h), and the square of its radius, (r). Calculate the percentage increase in its volume V when the radius increases by 5% and height h increases by 10%.

Step 1: Andika fomula ya ujazo wa awali. Ujazo V unabadilika kwa pamoja na urefu wake h na mraba wa eneo lake r. Hii inamaanisha kuna kigezo cha uwiano k. V_1 = k h_1 r_1^2 Ambapo V_1 ni ujazo wa awali, h_1 ni urefu wa awali, na r_1 ni eneo la awali. Step 2: Kokotoa eneo jipya na urefu mpya. Eneo r linaongezeka kwa 5%, hivyo eneo jipya r_2 ni: r_2 = r_1 + 0.05r_1 = 1.05r_1 Urefu h unaongezeka kwa 10%, hivyo urefu mpya h_2 ni: h_2 = h_1 + 0.10h_1 = 1.10h_1 Step 3: Kokotoa ujazo mpya V_2. Tumia fomula ya ujazo na maadili mapya ya h_2 na r_2: V_2 = k h_2 r_2^2 V_2 = k (1.10h_1) (1.05r_1)^2 V_2 = k (1.10h_1) (1.05^2 r_1^2) V_2 = k (1.10h_1) (1.1025 r_1^2) V_2 = (1.10 × 1.1025) k h_1 r_1^2 V_2 = 1.21275 k h_1 r_1^2 Kutoka Step 1, tunajua V_1 = k h_1 r_1^2. Hivyo, V_2 = 1.21275 V_1 Step 4: Kokotoa asilimia ya ongezeko la ujazo. Asilimia ya ongezeko inapatikana kwa fomula: (V_2 - V_1)/(V_1) × 100\%. Ongezeko la asilimia = (1.21275 V_1 - V_1)/(V_1) × 100\% Ongezeko la asilimia = ((1.21275 - 1)V_1)/(V_1) × 100\% Ongezeko la asilimia = 0.21275 × 100\% Ongezeko la asilimia = 21.275\% Asilimia ya ongezeko katika ujazo ni 21.275%. Ongezeko la asilimia = 21.275\% Tuma swali linalofuata 📸