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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Let's knock this out. 8. (a) A cylindrical metal pipe 3.5 m long has internal and external radii 5.0 cm and 7.5 cm respectively. Calculate the volume of: Given: Length of pipe (h) = 3.5 m = 3.5 × 100 cm = 350 cm Internal radius (r_i) = 5.0 cm External radius (r_e) = 7.5 cm Use = 3.14 (as specified in Q9, implying consistency for the paper). Step 1: Calculate the volume of water that can fill the pipe. The volume of water is the internal volume of the cylinder. V_water = r_i^2 h Substitute the values: V_water = 3.14 × (5.0 cm)^2 × (350 cm) V_water = 3.14 × 25 cm^2 × 350 cm V_water = 3.14 × 8750 cm^3 V_water = 27475 cm^3 Convert the volume from cm³ to litres (1 L = 1000 cm^3): V_water = (27475)/(1000) L V_water = 27.475 L Step 2: Calculate the volume of metal in the pipe. The volume of metal is the difference between the external volume and the internal volume of the pipe. V_metal = r_e^2 h - r_i^2 h = h (r_e^2 - r_i^2) Substitute the values: V_metal = 3.14 × 350 cm × ((7.5 cm)^2 - (5.0 cm)^2) V_metal = 3.14 × 350 cm × (56.25 cm^2 - 25 cm^2) V_metal = 3.14 × 350 cm × 31.25 cm^2 V_metal = 3.14 × 10937.5 cm^3 V_metal = 34361.25 cm^3 8. (b) A sector of a circle with radius 12 cm and angle 240° is used to form a cone. Calculate the: Given: Radius of sector (R) = 12 cm Angle of sector () = 240^ Step 3: Calculate the radius of the cone. When a sector is formed into a cone, the radius of the sector (R) becomes the slant height (l) of the cone, and the arc length of the sector becomes the circumference of the base of the cone. Arc length of sector = ()/(360^) × 2 R Circumference of cone base = 2 r, where r is the radius of the cone. Equate the two: 2 r = (240^)/(360^) × 2 (12 cm) 2 r = (2)/(3) × 24 cm 2 r = 16 cm Divide both sides by 2: r = (16)/(2) cm r = 8 cm Step 4: Calculate the vertical angle of the cone, correct to two decimal places. Let the vertical angle of the cone be . We have the radius of the cone (r = 8 cm) and the slant height (l = 12 cm). Consider the right-angled triangle formed by the cone's height, radius, and slant height. The angle at the apex of this triangle is ()/(2). (()/(2)) = oppositehypotenuse = (r)/(l) Substitute the values: (()/(2)) = 8 cm12 cm (()/(2)) = (2)/(3) Now, find ()/(2): ()/(2) = ((2)/(3)) ()/(2) ≈ 41.8103^ Multiply by 2 to find : ≈ 2 × 41.8103^ ≈ 83.6206^ Correct to two decimal places: = 83.62^ Send me the next one 📸