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: 12. Factorize completely: x^2 - 2xy^2 + y^4 Step 1: Recognize the pattern. The expression x^2 - 2xy^2 + y^4 is in the form of a perfect square trinomial, a^2 - 2ab + b^2 = (a-b)^2. Step 2: Identify 'a' and 'b'. In this case, a = x and b = y^2. Step 3: Substitute 'a' and 'b' into the perfect square formula. x^2 - 2x(y^2) + (y^2)^2 = (x - y^2)^2 The completely factorized expression is: (x - y^2)^2 13. Solve for y given that y is acute and (3y - 50^) = (2y + 10^) Step 1: Use the trigonometric identity = (90^ - ). We can rewrite the right side of the equation: (2y + 10^) = (90^ - (2y + 10^)) (2y + 10^) = (90^ - 2y - 10^) (2y + 10^) = (80^ - 2y) Step 2: Set the arguments of the sine functions equal. Now the equation is (3y - 50^) = (80^ - 2y). For A = B, we have two general cases: Case 1: A = B + 360^ n Case 2: A = 180^ - B + 360^ n Since y is acute, we will consider the primary solutions. Step 3: Solve for y using Case 1. 3y - 50^ = 80^ - 2y 3y + 2y = 80^ + 50^ 5y = 130^ y = (130^)/(5) y = 26^ This value is acute (0^ < 26^ < 90^). Step 4: Solve for y using Case 2. 3y - 50^ = 180^ - (80^ - 2y) 3y - 50^ = 180^ - 80^ + 2y 3y - 50^ = 100^ + 2y 3y - 2y = 100^ + 50^ y = 150^ This value is not acute. Therefore, the acute value for y is: y = 26^ 14. A solid consists of a cone and a hemisphere. The common diameter of the cone and the hemisphere is 10 cm, and the slanting height of the cone is 12 cm. Calculate correct to two decimal places the surface area of the solid. Step 1: Identify the dimensions. Common diameter = 10 cm. Radius (r) = 10 cm2 = 5 cm. Slanting height of the cone (l) = 12 cm. Step 2: Determine the components of the total surface area. The solid is formed by joining a cone and a hemisphere at their circular bases. Therefore, the total surface area of the solid is the sum of the curved surface area of the cone and the curved surface area of the hemisphere. Total Surface Area = Curved Surface Area of Cone + Curved Surface Area of Hemisphere. Step 3: Write down the formulas for each component. Curved Surface Area of Cone = r l Curved Surface Area of Hemisphere = 2 r^2 Step 4: Substitute the values into the formulas. Total Surface Area = (5 cm)(12 cm) + 2 (5 cm)^2 Total Surface Area = 60 cm^2 + 2 (25 cm^2) Total Surface Area = 60 cm^2 + 50 cm^2 Total Surface Area = 110 cm^2 Step 5: Calculate the numerical value and round to two decimal places. Using ≈ 3.14159265... Total Surface Area = 110 × 3.14159265 Total Surface Area ≈ 345.5751915 cm^2 Rounding to two decimal places: Total Surface Area ≈ 345.58 cm^2 The surface area of the solid is: 345.58 cm^2 Last free one today — make it count tomorrow, or type /upgrade for unlimited.