Question

type the correct answer in the box. use numerals instead of words. the pyramid of the sun in mexico is a square pyramid. the base edge is approximately 220 meters, and the height is approximately 70 meters. what is the approximate surface area of the sides of the pyramid to the nearest thousand square meters? the lateral surface area is about box

Explanation:

Step1: Find the slant height of the pyramid

The slant height \( l \) of a square pyramid can be found using the Pythagorean theorem. The base edge is \( 220 \) meters, so half of the base edge is \( \frac{220}{2}=110 \) meters. The height \( h = 70 \) meters. Then \( l=\sqrt{110^{2}+70^{2}}=\sqrt{12100 + 4900}=\sqrt{17000}\approx130.38 \) meters.

Step2: Calculate the lateral surface area

The lateral surface area \( LSA \) of a square pyramid is given by the formula \( LSA = 2\times base\ edge\times slant\ height \) (since there are 4 triangular faces, each with area \( \frac{1}{2}\times base\ edge\times slant\ height \), so total is \( 4\times\frac{1}{2}\times base\ edge\times slant\ height=2\times base\ edge\times slant\ height \)). Substituting the values, \( LSA = 2\times220\times130.38 = 440\times130.38\approx57367.2 \) square meters. Rounding to the nearest thousand, we get \( 57000 \) (wait, no, let's recalculate the slant height more accurately. \( \sqrt{17000}=\sqrt{100\times170}=10\sqrt{170}\approx10\times13.038 = 130.38 \). Then \( 2\times220\times130.38=440\times130.38 = 440\times130+440\times0.38=57200+167.2 = 57367.2 \). Wait, maybe I made a mistake in the formula. The correct formula for lateral surface area of a square pyramid is \( LSA=\frac{1}{2}\times perimeter\ of\ base\times slant\ height \). The perimeter of the base is \( 4\times220 = 880 \) meters. Then \( LSA=\frac{1}{2}\times880\times130.38 = 440\times130.38\approx57367.2 \). Rounding to the nearest thousand, 57367.2 is approximately 57000? Wait, no, 57367.2 is closer to 57000 or 58000? Wait, 57367.2 - 57000 = 367.2, 58000 - 57367.2 = 632.8, so closer to 57000? Wait, maybe my slant height calculation is wrong. Let's recalculate slant height: \( l=\sqrt{(220/2)^2 + 70^2}=\sqrt{110^2 + 70^2}=\sqrt{12100 + 4900}=\sqrt{17000}\approx130.38 \). Wait, maybe the problem expects using a different approach? Wait, maybe I miscalculated the lateral surface area. Wait, the lateral surface area of a square pyramid is 4 times the area of one triangular face. The area of one triangular face is \( \frac{1}{2}\times base\times slant\ height \). So \( 4\times\frac{1}{2}\times220\times130.38 = 2\times220\times130.38 = 440\times130.38\approx57367 \). Rounding to the nearest thousand, that's 57000? Wait, but maybe I made a mistake in the slant height. Wait, let's check again. \( 110^2 = 12100 \), \( 70^2 = 4900 \), sum is 17000, square root of 17000 is approximately 130.38. Then 440*130.38: 440*130 = 57200, 440*0.38 = 167.2, total 57367.2. So to the nearest thousand, that's 57000? Wait, but maybe the answer is 29000? No, that can't be. Wait, maybe I messed up the formula. Wait, no, the lateral surface area of a square pyramid is \( LSA = 2bs \), where \( b \) is the base edge and \( s \) is the slant height. Wait, no, the correct formula is \( LSA=\frac{1}{2}Pl \), where \( P \) is the perimeter of the base and \( l \) is the slant height. So \( P = 4*220 = 880 \), \( l=\sqrt{(220/2)^2 + 70^2}=\sqrt{110^2 + 70^2}=\sqrt{17000}\approx130.38 \). Then \( LSA = 0.5*880*130.38 = 440*130.38\approx57367 \), which is approximately 57000 when rounded to the nearest thousand. Wait, but maybe the problem expects a different approximation. Let's check with slant height approximated as 130. Then \( 440*130 = 57200 \), which is still ~57000. Alternatively, maybe I made a mistake in the slant height. Wait, 110-70-? Wait, no, the slant height is the height of the triangular face, so it's the hypotenuse of a right triangle with legs 110 (half of base) and 70 (height of pyramid). So that's correct. So the lateral surface area is approximately 57000? Wait, but let's check with another approach. Wait, maybe the answer is 29000? No, that would be if we used 220*70/2*4, but no, that's not correct. Wait, no, the area of a triangle is 1/2*base*height (slant height). So 4 triangles: 4*(1/2*220*130.38) = 2*220*130.38 = 440*130.38≈57367, which is ~57000. So I think the answer is 57000. Wait, but maybe the problem expects 29000? No, that can't be. Wait, maybe I miscalculated the slant height. Wait, 110 squared is 12100, 70 squared is 4900, sum is 17000, square root of 17000 is about 130.38, that's correct. So the lateral surface area is approximately 57000.

Answer:

57000