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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Here are the solutions to the problems from the image: 19. (c) A cone is made from the sector by joining OA to OB. The sector has a radius of 8 cm and a sector angle of 165° (from the previous parts of question 19). When the sector is formed into a cone: The radius of the sector (8 cm) becomes the slant height (l) of the cone. So, l = 8 cm. The arc length of the sector becomes the circumference of the base of the cone. (i) Calculate the radius, r, of the cone. Step 1: Recall the arc length of the sector. From part (a), the arc length L of the sector was calculated as: L = (22)/(3) cm Step 2: Equate the arc length to the circumference of the cone's base. The circumference of the cone's base is C = 2 r. 2 r = (22)/(3) Step 3: Solve for r. Divide both sides by 2: r = (22)/(3) ÷ (2) r = (22)/(3) × (1)/(2) r = (11)/(3) r ≈ 3.666... cm Rounding to two decimal places: r ≈ 3.67 cm The radius of the cone is 3.67 cm. (ii) Calculate the volume of the cone. [The volume, V, of a cone with radius r and height h is V = (1)/(3) r^2 h] Step 1: Find the height, h, of the cone. The slant height (l), radius (r), and height (h) of a cone form a right-angled triangle, so l^2 = r^2 + h^2. We have l = 8 cm and r = (11)/(3) cm. 8^2 = ((11)/(3))^2 + h^2 64 = (121)/(9) + h^2 h^2 = 64 - (121)/(9) h^2 = (576)/(9) - (121)/(9) h^2 = (455)/(9) h = sqrt((455)/(9)) h = sqrt(455)3 cm h ≈ 7.1102 cm Step 2: Calculate the volume of the cone. Using the formula V = (1)/(3) r^2 h: V = (1)/(3) ((11)/(3))^2 (sqrt(455)3) V = (1)/(3) ((121)/(9)) (sqrt(455)3) V = 121(455)81 V ≈ (121 × 3.14159265 × 21.3307125)/(81) V ≈ (8100.27)/(81) V ≈ 100.003 cm^3 Rounding to three significant figures: V ≈ 100 cm^3 The volume of the cone is 100 cm^3. Send me the next one 📸