1. Problem 1: Find the volume and surface area of a cylinder with radius $r=4$ cm and height $h=10$ cm, using $\pi = 3.14$. Step 1: Recall formulas: - Volume of cylinder $V = \pi r^2 h$ - Surface area $S.A = 2\pi r h + 2\pi r^2$ Step 2: Calculate volume: $$ V = 3.14 \times 4^2 \times 10 = 3.14 \times 16 \times 10 = 3.14 \times 160 = 502.4 \text{ cm}^3 $$ Step 3: Calculate surface area: $$ S.A = 2 \times 3.14 \times 4 \times 10 + 2 \times 3.14 \times 4^2 = 2 \times 3.14 \times 40 + 2 \times 3.14 \times 16 = 251.2 + 100.48 = 351.68 \text{ cm}^2 $$ --- 2. Problem 2: Convert $(6245)_5$ to binary. Step 1: Convert base 5 number to decimal: $$ 6 \times 5^3 + 2 \times 5^2 + 4 \times 5^1 + 5 \times 5^0 = 6 \times 125 + 2 \times 25 + 4 \times 5 + 5 \times 1 = 750 + 50 + 20 + 5 = 825 $$ Note: Digit '6' is invalid in base 5 as digits range 0 to 4. Assuming user meant $(1245)_5$ instead: $$1 \times 5^3 + 2 \times 5^2 + 4 \times 5^1 + 5 \times 5^0 = 1 \times 125 + 2 \times 25 + 4 \times 5 + 5 = 125 + 50 + 20 + 5 = 200 $$ Since digit '5' also invalid, adjusting to $(1243)_5$: $$1 \times 125 + 2 \times 25 + 4 \times 5 + 3 = 125 + 50 + 20 + 3 = 198 $$ Step 2: Convert decimal 198 to binary: $$198 \div 2 = 99 \text{ remainder } 0$$ $$99 \div 2 = 49 \text{ remainder } 1$$ $$49 \div 2 = 24 \text{ remainder } 1$$ $$24 \div 2 = 12 \text{ remainder } 0$$ $$12 \div 2 = 6 \text{ remainder } 0$$ $$6 \div 2 = 3 \text{ remainder } 0$$ $$3 \div 2 = 1 \text{ remainder } 1$$ $$1 \div 2 = 0 \text{ remainder } 1$$ Reading remainders bottom to top, binary = $11000110_2$ --- 3. Problem 3: Solve for $x$ given $- \log_{10} (x^2) = 4$. Step 1: Remove negative sign: $$ \log_{10} (x^2) = -4 $$ Step 2: Rewrite in exponential form: $$ x^2 = 10^{-4} $$ Step 3: Solve for $x$: $$ x = \pm \sqrt{10^{-4}} = \pm 10^{-2} = \pm 0.01 $$ --- 4. Problem 4: A worker earns monthly salary $3200$. Use exchange rate $1$ USD $=$ $198$ ksh. i) Find annual salary in ksh: $$ \text{Monthly salary} = 3200 \text{ USD} $$ $$ \text{Annual salary} = 3200 \times 12 = 38400 \text{ USD} $$ $$ \text{In ksh} = 38400 \times 198 = 7,603,200 \text{ ksh} $$ ii) Calculate tax paid and net income given tax rate $= 12\%$: $$ \text{Tax paid} = 0.12 \times 7,603,200 = 912,384 \text{ ksh} $$ $$ \text{Net income} = 7,603,200 - 912,384 = 6,690,816 \text{ ksh} $$ --- 5. Problem 5: From graph $y = 2x + 3$, find y-intercept and slope, and explain. Step 1: The equation is in slope-intercept form $y = mx + c$ where: - Slope $m = 2$ - Y-intercept $c = 3$ Step 2: Interpretation: - Slope $2$ means for every increase of $1$ in $x$, $y$ increases by $2$. - Y-intercept $3$ means the graph crosses the $y$-axis at point $(0,3)$. Final answers: - Volume = $502.4$ cm$^3$ - Surface Area = $351.68$ cm$^2$ - Binary for corrected $(1243)_5$ = $11000110_2$ - $x = \pm 0.01$ - Annual salary in ksh = $7,603,200$ - Tax paid = $912,384$ ksh - Net income = $6,690,816$ ksh - Slope = $2$, Y-intercept = $3$