This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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10. Volume of water in a cylindrical tin Step 1: Calculate the radius of the cylindrical tin. The diameter is 13.2 cm. r = diameter2 = 13.2 cm2 = 6.6 cm The height of the tin is h = 20 cm. Step 2: Calculate the total volume of the cylindrical tin. The formula for the volume of a cylinder is V = r^2 h. V_total = (6.6 cm)^2 (20 cm) V_total = (43.56 cm^2) (20 cm) V_total = 871.2 cm^3 Step 3: Calculate the volume of water. The tin is half-filled with water. V_water = (1)/(2) V_total V_water = (1)/(2) (871.2 cm^3) V_water = 435.6 cm^3 The correct option is D. D. 435.6 cm^3 11. Angle subtended by an arc Step 1: Use the arc length formula to find the angle in radians. The formula for arc length is L = r, where L is the arc length, r is the radius, and is the angle in radians. Given L = 22 cm and r = 15 cm. 22 cm = (15 cm) = (22)/(15) radians Step 2: Convert the angle from radians to degrees. To convert radians to degrees, multiply by (180^)/(). We are given to take = (22)/(7). _degrees = (22)/(15) × (180^)/(22)7 _degrees = (22)/(15) × (180^ × 7)/(22) _degrees = (1)/(15) × 180^ × 7 _degrees = 12^ × 7 _degrees = 84^ The correct option is C. C. 84^ 12. Common difference of an Arithmetic Progression Step 1: Write down the formulas for the terms of an Arithmetic Progression (AP). The first term is a = 20. The n-th term of an AP is given by T_n = a + (n-1)d, where d is the common difference. The third term is T_3 = a + (3-1)d = a + 2d. The fifth term is T_5 = a + (5-1)d = a + 4d. Step 2: Set up the equation based on the given condition. We are given that the fifth term is four-fifth of the third term: T_5 = (4)/(5) T_3 a + 4d = (4)/(5)(a + 2d) Step 3: Substitute the value of a and solve for d. Substitute a = 20 into the equation: 20 + 4d = (4)/(5)(20 + 2d) Multiply both sides by 5 to eliminate the fraction: 5(20 + 4d) = 4(20 + 2d) 100 + 20d = 80 + 8d Subtract 8d from both sides and subtract 100 from both sides: 20d - 8d = 80 - 100 12d = -20 d = (-20)/(12) d = -(5)/(3) The correct option is D. D. -(5)/(3) 13. Rounding to three significant figures Step 1: Identify the first three significant figures in the number 6.02517. The significant figures are counted from the first non-zero digit. 1st significant figure: 6 2nd significant figure: 0 3rd significant figure: 2 Step 2: Look at the fourth significant figure to determine rounding. The fourth significant figure is 5. Since the fourth significant figure (5) is 5 or greater, we round up the third significant figure. Step 3: Round the number. Rounding up the 2 (the third significant figure) makes it 3. So, 6.02517 rounded to three significant figures is 6.03. The correct option is B. B. 6.03 3 done, 2 left today. You're making progress.