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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Morning Mina — let's get this done. Here's the solution to question 5: Given: Length of a side of the base of the square-based pyramid, a = 20 cm. Length of the slant edge, s = 26 cm. a) A square-based pyramid has one square base and four triangular lateral faces. Therefore, the total number of surfaces is 1 + 4 = 5 surfaces. b) To find the total surface area (TSA) of the pyramid, we need the base area and the lateral surface area. The formula for TSA is TSA = Base Area + Lateral Surface Area. Base Area = a^2. Lateral Surface Area = 2al, where l is the slant height. Step 1: Calculate the vertical height (h) of the pyramid. We use the relationship between the slant edge (s), vertical height (h), and half of the base diagonal. Alternatively, we can use the relationship s^2 = h^2 + (asqrt(2)2)^2 = h^2 + (a^2)/(2). s^2 = h^2 + (a^2)/(2) Substitute the given values: 26^2 = h^2 + (20^2)/(2) 676 = h^2 + (400)/(2) 676 = h^2 + 200 h^2 = 676 - 200 h^2 = 476 h = sqrt(476) cm Step 2: Calculate the slant height (l) of the pyramid. We use the relationship between the slant height (l), vertical height (h), and half of the base side length ((a)/(2)). l^2 = h^2 + ((a)/(2))^2 Substitute the values for h^2 and a: l^2 = 476 + ((20)/(2))^2 l^2 = 476 + 10^2 l^2 = 476 + 100 l^2 = 576 l = sqrt(576) l = 24 cm Step 3: Calculate the total surface area (TSA). TSA = a^2 + 2al TSA = 20^2 + 2 × 20 × 24 TSA = 400 + 960 TSA = 1360 cm^2 The total surface area of the pyramid is 1360 cm^2. c) To compare the vertical height (h) and slant height (l): Step 1: State the values of h and l. Vertical height h = sqrt(476) cm. Slant height l = 24 cm. Step 2: Compare the values. We know that 21^2 = 441 and 22^2 = 484. So, sqrt(476) is between 21 and 22. h ≈ 21.82 cm. Since 24 > 21.82, the slant height is greater than the vertical height. Comparison: The slant height (l = 24 cm) is greater than the vertical height (h = sqrt(476) ≈ 21.82 cm). What's next?