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%.

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.

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:

$$h_2 = h_1 + 0.10h_1 = 1.10h_1$$

Tumia fomula ya ujazo na maadili mapya ya $h_2$ na $r_2$:

$$V_2 = k h_2 r_2^2$$ $$V_2 = k (1.10h_1) (1.05r_1)^2$$ $$V_2 = k (1.10h_1) (1.05^2 r_1^2)$$ $$V_2 = k (1.10h_1) (1.1025 r_1^2)$$ $$V_2 = (1.10 \times 1.1025) k h_1 r_1^2$$ $$V_2 = 1.21275 k h_1 r_1^2$$

Kutoka Step 1, tunajua $V_1 = k h_1 r_1^2$. Hivyo,

$$V_2 = 1.21275 V_1$$

Asilimia ya ongezeko inapatikana kwa fomula: $\frac{V_2 - V_1}{V_1} \times 100\%$.

$$\text{Ongezeko la asilimia} = \frac{1.21275 V_1 - V_1}{V_1} \times 100\%$$ $$\text{Ongezeko la asilimia} = \frac{(1.21275 - 1)V_1}{V_1} \times 100\%$$ $$\text{Ongezeko la asilimia} = 0.21275 \times 100\%$$ $$\text{Ongezeko la asilimia} = 21.275\%$$

Asilimia ya ongezeko katika ujazo ni 21.275%.