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
Here's how to find the volume of the pyramid: The problem asks for the volume of a regular pyramid. A regular pyramid has a regular polygon as its base and its apex is directly above the center of the base. Since the type of regular polygon for the base is not specified, we will assume it is a square pyramid, which is the most common interpretation in such problems. Given: • Side length of the base (s) = 9 cm • Slant height (l) = 18 cm The formula for the volume of a pyramid is: V = (1)/(3) × Base Area × Height Step 1: Calculate the area of the square base. The base is a square with side length s = 9 cm. Base Area (A_b) = s^2 A_b = 9^2 A_b = 81 cm^2 Step 2: Calculate the perpendicular height (h) of the pyramid. For a square base, the distance from the center of the base to the midpoint of a side (apothem, a) is half the side length. a = (s)/(2) a = (9)/(2) = 4.5 cm The slant height (l), the perpendicular height (h), and the apothem (a) form a right-angled triangle. We can use the Pythagorean theorem: l^2 = h^2 + a^2 18^2 = h^2 + (4.5)^2 324 = h^2 + 20.25 h^2 = 324 - 20.25 h^2 = 303.75 h = sqrt(303.75) To simplify sqrt(303.75): 303.75 = (30375)/(100) = (1215)/(4) h = sqrt((1215)/(4)) = sqrt(1215)sqrt(4) = sqrt(81 × 15)2 = 9sqrt(15)2 cm Step 3: Calculate the volume of the pyramid. V = (1)/(3) A_b h V = (1)/(3) (81) (9sqrt(15)2) V = 27 (9sqrt(15)2) V = 243sqrt(15)2 To get a numerical value: V ≈ (243 × 3.873)/(2) V ≈ (941.079)/(2) V ≈ 470.54 cm^3 The volume of the regular pyramid is 243sqrt(15)2 cm^3 ≈ 470.54 cm^3. That's 2 down. 3 left today — send the next one.