As always, multiply by two to account for the other side to get: $$3122.583545664$$. As mentioned above, a comes out to be around 16.2480768. The height of the side will be equal to a in the formula $a^2 + a^2 = 23^2$. To find the surface of C, the triangular prism's side, we will need to multiply the length, 96, by the height of the side. Again multiply by two to account for the other side resulting in: $$291.3333.$$. The surface area of B, the rectangular prism's width side, will also be simple as it is $23 * 6.33.$ which equals $145.66.$. Multiply that by two to account for the other side and get: $$1216$$. ft)įinding the surface area of A, the rectangular prism's length side, is as easy as $96*6.33.$ which equals $608$. ( The height of the pink area, d, is calculated using the total height of the structure, 12 ft, minus the height of the walls which was 6'4" or 6.33. The actual answer is that $a^2 = 264$ which means a is approximately $16.263455967$. I believe the issue in your solution is that you computate $a^2 = 132.25$. Thanks for the fun problem, and I hope that this helps! Go follow my Youtube Channel as I will be updating it in the future. Now the length of the rectangle, 96', times the height, 12.8203', is 1230.7526 * 2 (two sides) = 2461.5 ft^2įinal Answer! - Summing all of these parts together, we get A + B + C + D = 4099 There is now a right triangle with base of 11 1/2' and height of 5 2/3'. Divide the base by 2 to find the leg of the desired right triangle (with the slanted ceiling's side length being the hypotenuse of the aforementioned right triangle). The base of the isosceles triangle is 23', and the height is 5 2/3'. The length of these rectangles is 96' (as assumed in the OP), and we can find the height (triangle's shorter leg length) using the Gougu Theorem: This is perpendicular to the long side of the triangle, being 23' under the OP's assumption therefore, the area of the triangle is just (bh)/2 (not bh as shown in the OP). Height (altitude from vertex adjoined to equal side lengths) = 12' - 6 1/3' = 5 2/3'. This triangle is actually not a right triangle: Please Note: In the OP, you assumed that the triangle was a right triangle, then using the Gougu (Pythagorean) Theorem, not the Law of Cosines, to find the length of the legs of the isosceles triangle. Length: 23', Height, 6 1/3', Area = 23*(19/3) = 437/3 = 145.667 * 2 (two sides) = 291.333 ft^2Ĭ - Triangle sides of the triangular prism (assuming above the short side of the rectangle, as proposed in the OP) Moving on from this, let's solve for the four parts of this figure: (If you run into further issues, assumptions about dimensions may be withholding the correct answer) Please note that you did not specify in the original problem presentation the orientation of the triangular prism on top of the rectangular prism, so instead of assuming the triangles would be placed above the short sides of the green house, they could be above the longer sides, instead. D (2) Slanted ceiling surface rectangles.C (2) Triangle sides of the triangular prism.B (2) Short sides perpendicular to ground.A (2) Long sides perpendicular to ground.Here is what we must solve for in this problem now: With this same logic, the surface area of the floor should not be considered either. While there are a few ways to go about solving this problem, there are surely a few mistakes in the work shown here, but they can be quickly solved: We can assume that because it is a triangular prism on top of a rectangular prism that the roof will not be considered in the "outside" surface area of the Greenhouse (I'm following the idea that we will later use this surface area relative to the sunlight coming in through this amount of surface area, therefore only considering the outside surface area).
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