Calculate the percentage change in P, given that Q is increased by 10% while R is reduced by 20%

P inatofautiana kwa pamoja na Q na mraba wa R. Hii inamaanisha:

$$P = kQR^2$$

ambapo $k$ ni kigezo cha uwiano.

Tuseme maadili ya awali ni $P_1, Q_1, R_1$.
Hivyo, $P_1 = kQ_1R_1^2$.

Q imeongezeka kwa 10%, hivyo Q mpya ($Q_2$) ni:

$$Q_2 = Q_1 + 0.10Q_1 = 1.10Q_1$$

R imepungua kwa 20%, hivyo R mpya ($R_2$) ni:

$$R_2 = R_1 - 0.20R_1 = 0.80R_1$$

Badilisha $Q_2$ na $R_2$ kwenye mlinganyo wa uwiano:

$$P_2 = kQ_2R_2^2$$ $$P_2 = k(1.10Q_1)(0.80R_1)^2$$ $$P_2 = k(1.10Q_1)(0.64R_1^2)$$ $$P_2 = (1.10 \times 0.64) kQ_1R_1^2$$ $$P_2 = 0.704 kQ_1R_1^2$$

Kwa kuwa $P_1 = kQ_1R_1^2$, tunaweza kubadilisha $P_1$ kwenye mlinganyo wa $P_2$:

$$P_2 = 0.704 P_1$$

Asilimia ya mabadiliko inatolewa na fomula:

$$\text{Asilimia ya Mabadiliko} = \frac{P_2 - P_1}{P_1} \times 100\%$$ $$\text{Asilimia ya Mabadiliko} = \frac{0.704 P_1 - P_1}{P_1} \times 100\%$$ $$\text{Asilimia ya Mabadiliko} = (0.704 - 1) \times 100\%$$ $$\text{Asilimia ya Mabadiliko} = -0.296 \times 100\%$$ $$\text{Asilimia ya Mabadiliko} = -29.6\%$$

Asilimia hasi inaonyesha kupungua.

Asilimia ya mabadiliko katika P ni:

$$\boxed{\text{Kupungua kwa 29.6\%}}$$