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.
ScanSolve AI Answer
You're on a roll — Step 1: Find the surface area of the square-based pyramid. The pyramid has a square base and four triangular faces. Given: Side length of the square base (s) = 2sqrt(2) cm Slant height of the triangular faces (l) = 5 cm First, calculate the area of the square base (A_b): A_b = s^2 = (2sqrt(2))^2 = 2^2 × (sqrt(2))^2 = 4 × 2 = 8 cm^2 Next, calculate the area of one triangular face (A_t). The base of each triangle is the side length of the square base, s. A_t = (1)/(2) × base × slant height = (1)/(2) × s × l A_t = (1)/(2) × (2sqrt(2) cm) × (5 cm) A_t = (1)/(2) × 10sqrt(2) cm^2 = 5sqrt(2) cm^2 The total surface area (SA) is the sum of the base area and the area of the four triangular faces: SA = A_b + 4 × A_t SA = 8 cm^2 + 4 × (5sqrt(2) cm^2) SA = 8 cm^2 + 20sqrt(2) cm^2 SA = (8 + 20sqrt(2)) cm^2 (8 + 20sqrt(2)) cm^2 Step 2: Calculate the volume of the truncated cone (frustum). Given: Large radius (R) = 7 cm Small radius (r) = 5 cm Height of the frustum (H_f) = 10 cm Use = (22)/(7). The formula for the volume of a frustum is: V = (1)/(3) H_f (R^2 + Rr + r^2) Substitute the given values into the formula: V = (1)/(3) × (22)/(7) × 10 cm × ((7 cm)^2 + (7 cm × 5 cm) + (5 cm)^2) V = (220)/(21) cm × (49 cm^2 + 35 cm^2 + 25 cm^2) V = (220)/(21) cm × (109 cm^2) V = (220 × 109)/(21) cm^3 V = (23980)/(21) cm^3 (23980)/(21) cm^3 Got more? Send 'em!