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. Problem 10: Step 1: Find the prime factorization of each interval. The intervals are 9 minutes, 15 minutes, and 21 minutes. 9 = 3^2 15 = 3 × 5 21 = 3 × 7 Step 2: Calculate the Least Common Multiple (LCM) of the intervals. The LCM represents the time in minutes after which all bells will ring together again. LCM(9, 15, 21) = 3^2 × 5 × 7 = 9 × 5 × 7 = 315 minutes Step 3: Convert the LCM from minutes to hours and minutes. 315 minutes = (315)/(60) hours = 5 hours and 15 minutes Step 4: The bells will next ring together at 11:00 p.m. To find when they last rang together, subtract the LCM from 11:00 p.m. 11:00 p.m. - 5 hours 15 minutes 11:00 p.m. = 23:00 (in 24-hour format) 23:00 - 5 hours 15 minutes = 17:45 Converting back to 12-hour format: 17:45 = 5:45 p.m. The bells had last rang together at 5:45 p.m.. Problem 11: Step 1: Calculate the nominal area of the rectangle. Length L = 15 cm Width W = 10 cm Nominal Area A_0 = L × W = 15 cm × 10 cm = 150 cm^2. Step 2: Determine the maximum possible dimensions due to the 5% error. The error in each measurement is 5%. Maximum length L_max = L + (0.05 × L) = 15 cm + (0.05 × 15 cm) = 15 cm + 0.75 cm = 15.75 cm. Maximum width W_max = W + (0.05 × W) = 10 cm + (0.05 × 10 cm) = 10 cm + 0.5 cm = 10.5 cm. Step 3: Calculate the maximum possible area. A_max = L_max × W_max = 15.75 cm × 10.5 cm = 165.375 cm^2 Step 4: Calculate the maximum absolute error in the area. A_max = A_max - A_0 = 165.375 cm^2 - 150 cm^2 = 15.375 cm^2 Step 5: Calculate the maximum percentage error in the area. Percentage Error = A_maxA_0 × 100\% Percentage Error = 15.375 cm^2150 cm^2 × 100\% = 0.1025 × 100\% = 10.25\% The maximum percentage error in the area of the rectangle is 10.25\%. Problem 12: Step 1: Use the property that the sum of an interior angle and its corresponding exterior angle of a polygon is 180^. Given interior angle = 3x^ Given exterior angle = (x-20)^ 3x + (x-20) = 180 Step 2: Solve the equation for x. 4x - 20 = 180 4x = 180 + 20 4x = 200 x = (200)/(4) x = 50 Step 3: Calculate the measure of the exterior angle. Exterior angle = (x-20)^ = (50-20)^ = 30^. Step 4: Find the number of sides (n) of the polygon. For a regular polygon, the sum of the exterior angles is 360^, and each exterior angle is (360^)/(n). Exterior angle = (360^)/(n) 30^ = (360^)/(n) n = (360)/(30) n = 12 The number of sides of the polygon is 12. Send me the next one 📸