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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Step 1: Find the value of x. The sum of the interior angles of a quadrilateral is 360^. The given angles are 2(x-10), 4(x+5), 5(x+4), and (x-20). 2(x-10) + 4(x+5) + 5(x+4) + (x-20) = 360 Expand the terms: 2x - 20 + 4x + 20 + 5x + 20 + x - 20 = 360 Combine like terms: (2x + 4x + 5x + x) + (-20 + 20 + 20 - 20) = 360 12x + 0 = 360 12x = 360 Divide by 12: x = (360)/(12) x = 30 Step 2: Calculate the interior angles. Substitute x=30 into each angle expression: Angle A: 2(x-10) = 2(30-10) = 2(20) = 40^ Angle B: 4(x+5) = 4(30+5) = 4(35) = 140^ Angle C: 5(x+4) = 5(30+4) = 5(34) = 170^ Angle D: (x-20) = (30-20) = 10^ Step 3: Calculate the exterior angles. The sum of an interior angle and its corresponding exterior angle is 180^. Exterior Angle A: 180^ - 40^ = 140^ Exterior Angle B: 180^ - 140^ = 40^ Exterior Angle C: 180^ - 170^ = 10^ Exterior Angle D: 180^ - 10^ = 170^ The exterior angles of the quadrilateral are 140^, 40^, 10^, 170^. Step 1: Set up the work-rate relationship. This is a problem involving work, men, days, and hours. We can use the formula: WorkMen × Days × Hours = Constant Rate Let M_1, D_1, H_1, W_1 be the values for the first scenario and M_2, D_2, H_2, W_2 for the second scenario. Given: M_1 = 4 men W_1 = 2 acres D_1 = 3 days H_1 = 4 hours/day M_2 = ? men W_2 = 5 acres D_2 = 4 days H_2 = 3 hours/day Step 2: Formulate the equation. (W_1)/(M_1 D_1 H_1) = (W_2)/(M_2 D_2 H_2) Substitute the given values: (2)/(4 × 3 × 4) = (5)/(M_2 × 4 × 3) Step 3: Solve for M_2. (2)/(48) = (5)/(12 M_2) Simplify the left side: (1)/(24) = (5)/(12 M_2) Cross-multiply: 1 × (12 M_2) = 5 × 24 12 M_2 = 120 Divide by 12: M_2 = (120)/(12) M_2 = 10 Therefore, 10 men are required. Step 1: Determine the ratio of surface areas. For similar solids, the ratio of their surface areas is equal to the square of the ratio of their corresponding linear dimensions. Let A_s be the surface area of the smaller bottle and A_L be the surface area of the larger bottle. A_s = 12 cm^2 A_L = 108 cm^2 The ratio of surface areas is: (A_s)/(A_L) = (12)/(108) = (1)/(9) Step 2: Determine the ratio of linear dimensions. If the ratio of surface areas is k^2, then the ratio of linear dimensions is k. k^2 = (1)/(9) k = sqrt((1)/(9)) = (1)/(3) Step 3: Determine the ratio of volumes. For similar solids, the ratio of their volumes is equal to the cube of the ratio of their corresponding linear dimensions. (V_s)/(V_L) = k^3 (V_s)/(V_L) = ((1)/(3))^3 = (1^3)/(3^3) = (1)/(27) Step 4: Calculate the volume of the smaller bottle. We are given that the volume of the larger bottle (V_L) is 810 cm^3. (V_s)/(810 cm)^3 = (1)/(27) Multiply both sides by 810 cm^3: V_s = (1)/(27) × 810 cm^3 V_s = (810)/(27) cm^3 V_s = 30 cm^3 The volume of the smaller bottle is 30 cm^3. That's 2 down. 3 left today — send the next one.