Use logarithm tables to evaluate 4/(75.63)

Ili kujua ni lini mashine zitatuma ishara kwa wakati mmoja tena, tunahitaji kutafuta Kigawo Kidogo Zaidi cha Pamoja (LCM) cha vipindi vyao.

$$24 = 2^3 \times 3$$ $$27 = 3^3$$ $$30 = 2 \times 3 \times 5$$ $$50 = 2 \times 5^2$$

LCM ni zao la vigawo vikuu vyote vilivyoinuliwa kwa nguvu zao za juu zaidi.

$$\text{LCM}(24, 27, 30, 50) = 2^3 \times 3^3 \times 5^2$$ $$\text{LCM} = 8 \times 27 \times 25$$ $$\text{LCM} = 216 \times 25 = 5400 \text{ sekunde}$$

$$5400 \text{ sekunde} = \frac{5400}{60} \text{ dakika} = 90 \text{ dakika}$$ $$90 \text{ dakika} = 1 \text{ saa na } 30 \text{ dakika}$$

Mashine zilituma ishara kwa wakati mmoja saa 5:00 jioni.

$$\text{Muda unaofuata} = 5:00 \text{ jioni} + 1 \text{ saa } 30 \text{ dakika} = 6:30 \text{ jioni}$$

Mashine zitatuma ishara kwa wakati mmoja tena saa $\boxed{\text{6:30 jioni}}$.

11.

Acha $x = \sqrt[4]{\frac{37^2 \times 0.0168}{75.63}}$.

$$\log x = \log \left( \frac{37^2 \times 0.0168}{75.63} \right)^{\frac{1}{4}}$$ $$\log x = \frac{1}{4} [2 \log 37 + \log 0.0168 - \log 75.63]$$

Kutoka kwenye jedwali la logariti:

$$\log 37 = 1.5682$$ $$\log 0.0168 = \bar{2}.2253$$ $$\log 75.63 = 1.8787$$

$$2 \log 37 = 2 \times 1.5682 = 3.1364$$ $$\log(\text{nambari ya juu}) = 3.1364 + \bar{2}.2253 = 1.3617$$

$$\log \left(\frac{37^2 \times 0.0168}{75.63}\right) = 1.3617 - 1.8787$$ $$1.3617 - 1.8787 = -0.5170$$

Ili kuandika kwa tabia yenye bar:

$$-0.5170 = -1 + 0.4830 = \bar{1}.4830$$

$$\log x = \frac{1}{4} (\bar{1}.4830)$$

Ili kugawanya, tunabadilisha tabia ili iweze kugawanyika kwa 4:

$$\bar{1}.4830 = -4 + 3.4830$$ $$\log x = \frac{-4 + 3.4830}{4} = -1 + 0.87075$$ $$\log x = \bar{1}.8708 \quad (\text{kwa sehemu nne za desimali})$$

Kutoka kwenye jedwali la antilogariti, mantissa 0.8708 inatoa 7427. Tabia $\bar{1}$ inamaanisha kuna sifuri moja kabla ya desimali.

$$x = 0.7427$$

Thamani ya usemi ni $\boxed{0.7427}$.

12.

Tunajua msongamano ($\rho$) = Masi (m) / Ujazo (V). Hivyo, $V = \frac{m}{\rho}$.

$$V = \frac{63 \text{ Kg}}{7000 \text{ Kg/m}^3} = 0.009 \text{ m}^3$$

Vipimo vya nje: 12 cm na 15 cm. Badilisha kuwa mita:

$$12 \text{ cm} = 0.12 \text{ m}$$ $$15 \text{ cm} = 0.15 \text{ m}$$ $$\text{Eneo la nje} = 0.12 \text{ m} \times 0.15 \text{ m} = 0.018 \text{ m}^2$$

Vipimo vya ndani: 10 cm na 12 cm. Badilisha kuwa mita:

$$10 \text{ cm} = 0.10 \text{ m}$$ $$12 \text{ cm} = 0.12 \text{ m}$$ $$\text{Eneo la ndani} = 0.10 \text{ m} \times 0.12 \text{ m} = 0.012 \text{ m}^2$$

Eneo la chuma = Eneo la nje - Eneo la ndani

$$\text{Eneo la chuma} = 0.018 \text{ m}^2 - 0.012 \text{ m}^2 = 0.006 \text{ m}^2$$

Ujazo wa chuma = Eneo la chuma $\times$ Urefu wa bomba (L)

$$L = \frac{\text{Ujazo wa chuma}}{\text{Eneo la chuma}}$$ $$L = \frac{0.009 \text{ m}^3}{0.006 \text{ m}^2} = 1.5 \text{ m}$$

Urefu wa bomba ni $\boxed{1.5 \text{ m}}$.