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 questions 3, 8, 9, and 14. Question 3: a) Find the ratio of their volumes. b) If the volume of the larger cone is 25 cm^3, what is the volume of the smaller cone? Step 1: Determine the relationship between the ratio of radii and the ratio of volumes for similar cones. For similar figures, if the ratio of corresponding lengths (like radii) is a:b, then the ratio of their volumes is a^3:b^3. Given the ratio of radii is 2:5. Step 2: Calculate the ratio of their volumes. V_smallerV_larger = (r_smallerr_larger)^3 = ((2)/(5))^3 = (2^3)/(5^3) = (8)/(125) The ratio of their volumes is 8:125. Step 3: Calculate the volume of the smaller cone. Given the volume of the larger cone V_larger = 25 cm^3. V_smaller = (8)/(125) × V_larger = (8)/(125) × 25 cm^3 = (8 × 25)/(125) cm^3 = (200)/(125) cm^3 = (8)/(5) cm^3 V_smaller = 1.6 cm^3 a) The ratio of their volumes is 8:125. b) The volume of the smaller cone is 1.6 cm^3. Question 8: A tin full of melon with a radius of 1.5 cm costs N10. How much would a similar tin with radius 4.5 cm cost? Step 1: Determine the ratio of the radii of the two similar tins. Let r_1 be the radius of the first tin and r_2 be the radius of the second tin. (r_1)/(r_2) = 1.5 cm4.5 cm = (1)/(3) Step 2: Determine the ratio of their volumes. Since the cost is proportional to the volume of the melon, we need the ratio of volumes. For similar shapes, the ratio of volumes is the cube of the ratio of corresponding lengths (radii). (V_1)/(V_2) = ((r_1)/(r_2))^3 = ((1)/(3))^3 = (1^3)/(3^3) = (1)/(27) Step 3: Calculate the cost of the second tin. Let C_1 be the cost of the first tin and C_2 be the cost of the second tin. (C_1)/(C_2) = (V_1)/(V_2) N10C_2 = (1)/(27) C_2 = N10 × 27 = N270 The similar tin with radius 4.5 cm would cost N270. Question 9: A metal rod costs N250. Calculate the cost of a similar rod if the length is (2)/(5) of the first one. Step 1: Determine the ratio of the lengths of the two similar rods. Let L_1 be the length of the first rod and L_2 be the length of the second rod. Given L_2 = (2)/(5) L_1. (L_2)/(L_1) = (2)/(5) Step 2: Determine the ratio of their volumes. Assume the cost of the metal rod is proportional to its volume. For similar shapes, the ratio of volumes is the cube of the ratio of corresponding lengths. (V_2)/(V_1) = ((L_2)/(L_1))^3 = ((2)/(5))^3 = (2^3)/(5^3) = (8)/(125) Step 3: Calculate the cost of the second rod. Let C_1 be the cost of the first rod and C_2 be the cost of the second rod. (C_2)/(C_1) = (V_2)/(V_1) (C_2)/(N)250 = (8)/(125) C_2 = (8)/(125) × N250 = 8 × (250)/(125) = 8 × 2 = N16 The cost of the similar rod is N16. Question 14: Two similar shapes have volumes 27 cm^3 and 64 cm^3. If the surface area of the larger shape is 5.2 m^2, find the surface area of the smaller shape. Step 1: Determine the ratio of the volumes of the two shapes. Let V_S be the volume of the smaller shape and V_L be the volume of the larger shape. (V_S)/(V_L) = 27 cm^364 cm^3 = (27)/(64) Step 2: Determine the ratio of their corresponding lengths (scale factor). For similar figures, the ratio of volumes is the cube of the ratio of lengths. So, the ratio of lengths is the cube root of the ratio of volumes. (L_S)/(L_L) = 3/(V_L) = 3/(64) = [3]27[3]64 = (3)/(4) Step 3: Determine the ratio of their surface areas. For similar figures, the ratio of surface areas is the square of the ratio of corresponding lengths. (A_S)/(A_L) = ((L_S)/(L_L))^2 = ((3)/(4))^2 = (3^2)/(4^2) = (9)/(16) Step 4: Calculate the surface area of the smaller shape. Given the surface area of the larger shape A_L = 5.2 m^2. A_S = (9)/(16) × A_L = (9)/(16) × 5.2 m^2 A_S = (46.8)/(16) m^2 = 2.925 m^2 The surface area of the smaller shape is 2.925 m^2. Send me the next one 📸