P varies as the square of R. R varies as the square of T. When P=18, R=3 and T=5. Express P in terms of T hence find P when T=10.

P inatofautiana moja kwa moja na mraba wa R. Hii inaweza kuandikwa kama:

$$P = k_1 R^2$$

Ambapo $k_1$ ni kigezo cha uwiano.

R inatofautiana moja kwa moja na mraba wa T. Hii inaweza kuandikwa kama:

$$R = k_2 T^2$$

Ambapo $k_2$ ni kigezo cha uwiano.

Badilisha $R$ kutoka Hatua ya 2 kwenye mlinganyo wa Hatua ya 1:

$$P = k_1 (k_2 T^2)^2$$ $$P = k_1 k_2^2 T^4$$

Tunaweza kuweka $K = k_1 k_2^2$, hivyo mlinganyo unakuwa:

$$P = K T^4$$

Tunapewa $P = 18$, $R = 3$, na $T = 5$.
Kutoka $P = k_1 R^2$:

$$18 = k_1 (3)^2$$ $$18 = 9k_1$$ $$k_1 = \frac{18}{9} = 2$$

Kutoka $R = k_2 T^2$:

$$3 = k_2 (5)^2$$ $$3 = 25k_2$$ $$k_2 = \frac{3}{25}$$

Sasa, badilisha $k_1$ na $k_2$ kwenye mlinganyo wa P kwa kutumia T:

$$P = k_1 k_2^2 T^4$$ $$P = 2 \left(\frac{3}{25}\right)^2 T^4$$ $$P = 2 \left(\frac{9}{625}\right) T^4$$ $$P = \frac{18}{625} T^4$$

Huu ndio usemi wa P kwa kutumia T.

Tumia mlinganyo uliopatikana katika Hatua ya 4:

$$P = \frac{18}{625} (10)^4$$ $$P = \frac{18}{625} (10000)$$ $$P = 18 \times \frac{10000}{625}$$ $$P = 18 \times 16$$ $$P = 288$$

Thamani ya P wakati T = 10 ni $\boxed{\text{288}}$.