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: Define original dimensions and area. Let the original length of the rectangle be L and the original width be W. The original area, A_1, is given by: A_1 = L × W Step 2: Calculate new dimensions. The sides are increased by 10%. New length, L', is L + 0.10L = 1.10L. New width, W', is W + 0.10W = 1.10W. Step 3: Calculate the new area. The new area, A_2, is given by: A_2 = L' × W' = (1.10L) × (1.10W) = 1.21LW Step 4: Calculate the percentage change in area. The percentage change is given by the formula: Percentage Change = (A_2 - A_1)/(A_1) × 100\% Substitute the expressions for A_1 and A_2: Percentage Change = (1.21LW - LW)/(LW) × 100\% Percentage Change = (0.21LW)/(LW) × 100\% Percentage Change = 0.21 × 100\% Percentage Change = 21\% The percentage change in its area is 21%. 15. Step 1: Define the two-digit number. Let the two-digit number be 10t + u, where t is the tens digit and u is the units digit. Step 2: Formulate the first equation. "the sum of its digits is 9" t + u = 9 (Equation 1) Step 3: Formulate the second equation. "When the digits are reversed, the new number exceeds the old number by 45." The new number with digits reversed is 10u + t. The old number is 10t + u. So, the difference is: (10u + t) - (10t + u) = 45 10u + t - 10t - u = 45 9u - 9t = 45 Divide by 9: u - t = 5 (Equation 2) Step 4: Solve the system of equations. We have: 1) t + u = 9 2) -t + u = 5 Add Equation 1 and Equation 2: (t + u) + (-t + u) = 9 + 5 2u = 14 u = 7 Substitute u = 7 into Equation 1: t + 7 = 9 t = 9 - 7 t = 2 Step 5: Find the original number. The original number is 10t + u. 10(2) + 7 = 20 + 7 = 27 The original number is 27. 16. Step 1: Identify the formula for the volume of a prism. The volume of a prism is given by the area of its cross-section multiplied by its length. V = A_base × L Step 2: Calculate the area of the regular hexagonal cross-section. A regular hexagon can be divided into 6 equilateral triangles. The side length of the hexagon is s = 5 cm. The area of one equilateral triangle with side s is sqrt(3)4s^2. The area of the regular hexagonal base, A_base, is: A_base = 6 × sqrt(3)4s^2 A_base = 3sqrt(3)2s^2 Substitute s = 5 cm: A_base = 3sqrt(3)2(5)^2 A_base = 3sqrt(3)2(25) A_base = 75sqrt(3)2 cm^2 Step 3: Use the given length of the prism. The length of the prism, L, is 12 cm. Step 4: Calculate the volume of the prism. V = A_base × L V = 75sqrt(3)2 × 12 V = 75sqrt(3) × 6 V = 450sqrt(3) cm^3 The volume of the prism is 450sqrt(3) cm^3.