B varies partly as the square of M and partly as the inverse of H. Such that when M=2, H=1/2 B=76. While when sqrt(M)=3, H=2 B=46. Write an expression for B in term M and H.

B inatofautiana kwa sehemu kama mraba wa M ($M^2$) na kwa sehemu kama kinyume cha N ($\frac{1}{N}$). Hii inamaanisha tunaweza kuandika uhusiano kama:

$$B = k_1 M^2 + \frac{k_2}{N}$$

ambapo $k_1$ na $k_2$ ni vigezo vya uwiano.

Tunapewa $M=2$, $N=\frac{1}{2}$, na $B=76$. Badilisha hizi kwenye fomula:

$$76 = k_1 (2)^2 + \frac{k_2}{\frac{1}{2}}$$ $$76 = 4k_1 + 2k_2 \quad \text{(Mlinganyo 1)}$$

Tunapewa $\sqrt{M}=3$, $N=2$, na $B=46$.
Kutoka $\sqrt{M}=3$, tunapata $M = 3^2 = 9$.
Badilisha $M=9$, $N=2$, na $B=46$ kwenye fomula:

$$46 = k_1 (9)^2 + \frac{k_2}{2}$$ $$46 = 81k_1 + \frac{1}{2}k_2 \quad \text{(Mlinganyo 2)}$$

Kutoka Mlinganyo 1, gawanya kwa 2:

$$38 = 2k_1 + k_2 \implies k_2 = 38 - 2k_1$$

Badilisha $k_2$ kwenye Mlinganyo 2:

$$46 = 81k_1 + \frac{1}{2}(38 - 2k_1)$$ $$46 = 81k_1 + 19 - k_1$$ $$46 - 19 = 80k_1$$ $$27 = 80k_1$$ $$k_1 = \frac{27}{80}$$

Sasa tafuta $k_2$:

$$k_2 = 38 - 2\left(\frac{27}{80}\right)$$ $$k_2 = 38 - \frac{27}{40}$$ $$k_2 = \frac{38 \times 40 - 27}{40}$$ $$k_2 = \frac{1520 - 27}{40}$$ $$k_2 = \frac{1493}{40}$$

Badilisha thamani za $k_1$ na $k_2$ kwenye fomula ya awali:

$$B = \frac{27}{80} M^2 + \frac{1493}{40N}$$