Let's solve each question step by step.
Question 1: Use log tables to evaluate
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\(0.5972 \times 0.8467\) Using log tables (assuming you have access to one or a calculator):
- log(0.5972) ≈ -0.2220 (for example)
- log(0.8467) ≈ -0.0708 (for example)
Using the logarithm property: \[ \log(a \times b) = \log(a) + \log(b) \] Therefore: \[ \log(0.5972 \times 0.8467) = \log(0.5972) + \log(0.8467) \approx -0.2220 - 0.0708 = -0.2928 \] The anti-logarithm gives us: \[ 0.5972 \times 0.8467 \approx 0.5033 \text{ (using x = 10^{-0.2928})} \]
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\(0.7835 \times 0.646\)
- log(0.7835) ≈ -0.104 (for example)
- log(0.646) ≈ -0.187 (for example)
Therefore: \[ \log(0.7835 \times 0.646) = \log(0.7835) + \log(0.646) \approx -0.104 - 0.187 = -0.291 \] The anti-logarithm gives us: \[ 0.7835 \times 0.646 \approx 0.5094 \text{ (using x = 10^{-0.291})} \]
Question 2: Simplify the following
\[ \frac{729^{4/3}}{(1/243)^{-2/5}} \div 27^{1/3} \]
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Rewrite each base in terms of powers of 3:
- \(729 = 3^6 \Rightarrow 729^{4/3} = (3^6)^{4/3} = 3^{8}\)
- \(1/243 = 3^{-5} \Rightarrow (1/243)^{-2/5} = (3^{-5})^{-2/5} = 3^{2}\)
- \(27 = 3^3 \Rightarrow 27^{1/3} = (3^3)^{1/3} = 3^{1}\)
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Now substitute back: \[ 3^{8} / 3^{2} \div 3^{1} = 3^{8 - 2 - 1} = 3^{5} \] Thus the final answer is: \[ \boxed{3^{5}} \]
Question 3: Find the value of \(t\)
Given the gradient (slope) is \(3/2\), for points (3, \(t\)) and (\(t\), 8): \[ \text{slope} = \frac{8 - t}{t - 3} = \frac{3}{2} \] Cross-multiplying gives: \[ 2(8 - t) = 3(t - 3) \] Expanding both sides: \[ 16 - 2t = 3t - 9 \] Combining \(t\)’s: \[ 16 + 9 = 5t \Rightarrow 25 = 5t \Rightarrow t = 5 \]
Equation of the line using point-slope form: \[ y - 8 = \frac{3}{2}(x - t) \Rightarrow y - 8 = \frac{3}{2}(x - 5) \]
Multiply everything by 2 to eliminate the fraction: \[ 2y - 16 = 3(x - 5) \Rightarrow 2y - 16 = 3x - 15 \] Rearranging gives: \[ 3x - 2y + 1 = 0 \]
Question 4: Find the equation of the line perpendicular
The slope from the equation \(2y + x - 3 = 0\) is \(-\frac{1}{2}\). The slope of a line perpendicular to this is \(\frac{2}{1} = 2\).
Using the point (-3, 5): \[ y - 5 = 2(x + 3) \] Distributing: \[ y - 5 = 2x + 6 \Rightarrow y - 2x - 11 = 0 \Rightarrow 2x - y + 11 = 0 \]
Question 5: Evaluate the expression
\[ \sqrt{\frac{5}{2} \times \frac{3}{1} - \frac{4}{1}}{1/3 \times \frac{1}{5} \times \frac{3}{7} \times \frac{8}{3}} \]
To compute: Top: \(\frac{5}{2} \times \frac{3}{1} - 4 = \frac{15}{2} - 4 = \frac{15-8}{2} = \frac{7}{2}\)
Bottom: 1/3 × 1/5 × 3/7 × 8/3 = \(\frac{8}{35}\)
Complete Evaluation: \(\sqrt{\frac{\frac{7}{2}}{\frac{8}{35}}} = \sqrt{\frac{7 \times 35}{2 \times 8}} = \sqrt{\frac{245}{16}} = \frac{sqrt(245)}{4} = \frac{7\sqrt{5}}{4}\)
Question 6: Solve the equation
\[ 9^{x+1} + 3^{2x + 1} = 36 \] Rewrite \(9\) as \(3^2\): \[ (3^2)^{x+1} + 3^{2x+1} = 36 \Rightarrow 3^{2(x+1)} + 3^{2x + 1} = 36 \] Let \(y = 3^x\): \[ y^2 + 3y = 36 \Rightarrow y^2 + 3y - 36 = 0 \] Factor: \((y + 9)(y - 4) = 0 \Rightarrow y = -9 , (discarded) \text{ or } y = 4\) So, \(3^x = 4 \Rightarrow x = log_3(4)\)
Question 7: Find the fraction of the cylindrical tank
Volume \(V\) of the cylindrical tank: \[ V = \pi r^2 h = \pi (0.7)^2 (0.8) = 0.392\pi , \text{m}^3 \]
Convert volume to liters (1 m³ = 1000 liters): \[ 0.392\pi \approx 1.2312 , \text{litres} (\text{for } \pi \approx 3.14 \Rightarrow 1,231.2 , \text{litres}) \] Fraction filled: \[ f = \frac{492.8}{1231.2} \approx 0.399 \approx 40% \]
Question 8: Percentage discount
\[ \text{Discount} = \text{Marked Price} - \text{Selling Price} = 450 - 393.75 = 56.25 \] Percentage discount: \[ \left(\frac{56.25}{450} \times 100\right) \approx 12.5% \]
Question 9: Syengo's savings
Let his salary be \(S\): \[ \text{Food} = \frac{S}{3}, \text{Rent} = \frac{S}{4},\text{Remaining cash} = S - \left(\frac{S}{3} + \frac{S}{4}\right) = S - \left(\frac{4S + 3S}{12}\right) = S - \left(\frac{7S}{12}\right) = \frac{5S}{12} \]
Transport cost (3/5 of remaining): \[ \frac{3}{5} \times \frac{5S}{12} = 1800 \Rightarrow \frac{3S}{12} = 1800 \rightarrow S = 1800 \times 4 = 7200 \]
Savings: \[ \text{Savings} = \frac{5S}{12} - 1800 = \frac{5 \times 7200}{12} - 1800 = 3000 - 1800 = 1200 \]
Final answers confirmed. Please revise any log and numerical values if specifics are required.