If the height of a pyramid is 15cm, the side length of the bottom is 12cm, and the side area of the inscribed pyramid is 120cm2, (1) find the height of the pyramid; (2) find the ratio of the side area of the small pyramid of the bottom section of the pyramid to that of the original pyramid

If the height of a pyramid is 15cm, the side length of the bottom is 12cm, and the side area of the inscribed pyramid is 120cm2, (1) find the height of the pyramid; (2) find the ratio of the side area of the small pyramid of the bottom section of the pyramid to that of the original pyramid

(1) Let the height of the inscribed regular triangular prism be x, and the side length of its bottom be a. similar to the right triangle, we can get 15 − x15 = 23 × 32a23 × 32 × 12, ∧ a = 60 − 4x5, and the side area of the inscribed regular triangular prism is 120 = 3A · x = 360 − 4x5 & nbsp; X, x2-15x + 50 = 0, ∧ x = 10 & nbsp; (2) the ratio of the side areas of two pyramids is equal to the square of the corresponding similarity ratio, and the similarity ratio is 15 & nbsp; − x15, when x = 10 & nbsp; When x = 5, the similarity ratio is 23, so the ratio of the side area of the two pyramids is 49, so the ratio of the side area of the small pyramid on the bottom section of the prism to the original pyramid is 1:9 or 4:9

In a regular triangular pyramid p-abc, the length of the side edge is 10 cm, the angle between the side edge and the bottom is α, cos α = 4 / 5, the height, the area and volume of the side edge are calculated

(draw a picture on the sketch paper and compare with my steps)
Make Po ⊥ plane ABC perpendicular foot o, then o is triangle ABC center
Cos α = AO / PA = 4 / 5 because PA = 10
So Po = 6 Ao = 8 (AO is the radius of circumcircle of triangle ABC denoted by R)
According to the sine theorem BC / Sina = 2R, we can get that:
BC = 2R * Sina = 16 * sin60 ° = 8 √ 3, that is, the side length of the bottom triangle is 8 √ 3
Let PE ⊥ AB be e. from Pythagorean theorem PE = √ (PA & # 178; - AE & # 178;) = 2 √ 13
S side = (1 / 2) AB * PE * 3 = 24 √ 39
V = (1 / 3) s bottom * Po = (1 / 3) * 48 √ 3 * 6 = 96 √ 3

As shown in the figure, the length of the bottom edge BC of △ ABC is 10cm. When vertex a moves upward from point D on the vertical line PD of BC, the area of the triangle changes. (1) in the process of this change, the independent variable is______ The dependent variable is______ (2) if ad is xcm and area is ycm2, it can be expressed as y=______ (3) when ad = BC, the area of △ ABC is______ ．

(1) In the process of this change, the independent variable is the height of the triangle, and the dependent variable is the area of the triangle; (2) if ad is xcm and the area is ycm2, it can be expressed as y = 5xcm2; (3) when ad = BC, the area of △ ABC is 50cm2; so the answer is: the height of the triangle, the area of the triangle; 5xcm2; 50cm2