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.
ScanSolve AI Answer
Welcome back Samson — missed you this week. Here's the solution to question 8. Step 1: Determine the ratio of the surface areas. Let A_1 be the surface area of the smaller bottle and A_2 be the surface area of the larger bottle. Given A_1 = 12 cm^2 and A_2 = 108 cm^2. The ratio of the surface areas is: (A_1)/(A_2) = 12 cm^2108 cm^2 = (1)/(9) Step 2: Find the ratio of the corresponding linear dimensions (scale factor). For similar objects, the ratio of their surface areas is equal to the square of the ratio of their corresponding linear dimensions (scale factor, k). (A_1)/(A_2) = k^2 (1)/(9) = k^2 To find k, take the square root of both sides: k = sqrt((1)/(9)) = (1)/(3) So, the ratio of the linear dimensions of the smaller bottle to the larger bottle is (1)/(3). Step 3: Calculate the volume of the smaller bottle. For similar objects, the ratio of their volumes is equal to the cube of the ratio of their corresponding linear dimensions. Let V_1 be the volume of the smaller bottle and V_2 be the volume of the larger bottle. Given V_2 = 810 cm^3. (V_1)/(V_2) = k^3 (V_1)/(810 cm)^3 = ((1)/(3))^3 (V_1)/(810 cm)^3 = (1)/(27) Now, solve for V_1: V_1 = (1)/(27) × 810 cm^3 V_1 = (810)/(27) cm^3 V_1 = 30 cm^3 The volume of the smaller bottle is 30 cm^3. Send me the next one 📸