Example Question - surface area calculation
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Calculating Surface Area of Composite 3D Figure
To find the surface area of the 3D figure provided in the image, you need to calculate the area of each visible surface and then add them all together. The figure is a composite solid made up of two rectangular prisms. We must be careful not to double-count any area that is shared between the two prisms. 1. The large, vertical front face is 16 in. by 16 in., so its area is \(16\ in \times 16\ in = 256\ in^2\). 2. The top face of the larger prism is 16 in. by 8 in., so its area is \(16\ in \times 8\ in = 128\ in^2\). 3. The back face of the larger prism, which is identical to the front, is also \(16\ in \times 16\ in = 256\ in^2\). 4. The bottom face of the larger prism is identical to the top face and has an area of \(16\ in \times 8\ in = 128\ in^2\). 5. The left face of the large prism is 16 in. by 8 in., giving an area of \(16\ in \times 8\ in = 128\ in^2\). Now let's consider the smaller, attached prism: 6. The visible part of the right side of the larger prism is an 8 in. by 8 in. square (above the smaller prism), with an area of \(8\ in \times 8\ in = 64\ in^2\). 7. The front face of the smaller prism is 6 in. by 8 in., giving an area of \(6\ in \times 8\ in = 48\ in^2\). 8. The top face of the smaller prism (visible portion) is 6 in. by 8 in., with an area of \(6\ in \times 8\ in = 48\ in^2\). 9. The right-side face of the smaller prism (aligned with the front face of the larger prism) is 6 in. by 16 in., with an area of \(6\ in \times 16\ in = 96\ in^2\). Now let's add up all these areas to find the total surface area: \(256\ in^2 + 128\ in^2 + 256\ in^2 + 128\ in^2 + 128\ in^2 + 64\ in^2 + 48\ in^2 + 48\ in^2 + 96\ in^2 = 1152\ in^2\). So the total surface area of the figure is 1,152 square inches.
Counting Visible Cubes on a Composite Cube
Die Übung bezieht sich darauf, die Anzahl der kleinen Würfel in einem größeren Würfel (wie bei einem Rubik's Cube) zu bestimmen, wobei die inneren Würfel (die nicht sichtbar sind) nicht gezählt werden sollen. In diesem Fall haben wir einen Würfel, der so aussieht wie ein Rubik's Cube und scheint aus kleineren Würfeln zusammengesetzt zu sein. Um die Anzahl der sichtbaren kleineren Würfel zu bestimmen, müssen wir die Anzahl der Würfel auf jeder Seite des größeren Würfels zählen und dann die Würfel an den Kanten und Ecken so adjustieren, dass keine doppelt gezählt werden. Hier ist eine allgemeine Formel, um die Anzahl der sichtbaren kleineren Würfel auf einem Würfel mit der Kantenlänge n zu berechnen: Anzahl der sichtbaren kleineren Würfel = \( 6n^2 - 12n + 8 \) Die Begründung für diese Formel ist folgendermaßen: - \( 6n^2 \): Jede der 6 Seitenflächen des Würfels hat \( n^2 \) kleine Würfel. - \( -12n \): Die Kanten haben doppelt gezählte Würfel. Da es 12 Kanten gibt und jede Kante \( n \) Würfel hat (die Ecken nicht doppelt zählen), subtrahieren wir \( 12n \). - \( +8 \): Schlussendlich addieren wir 8 hinzu für die acht Ecken, da jede Ecke in der ursprünglichen Rechnung 3-mal abgezogen wurde und 1-mal gezählt werden soll. Wenn der Würfel die Kantenlänge 3 hat (wie beim typischen Rubik's Cube), setzen wir in die Formel n=3 ein: Anzahl der sichtbaren kleineren Würfel = \( 6*3^2 - 12*3 + 8 \) = \( 6*9 - 36 + 8 \) = \( 54 - 36 + 8 \) = \( 26 \) Es gibt also 26 sichtbare kleinere Würfel an der Oberfläche eines Würfels mit einer Kantenlänge von 3.
Calculating Required Paint Cans for Ceiling
Để giải quyết bài toán, ta cần tính diện tích của trần nhà hình chữ nhật, sau đó chia diện tích đó cho diện tích mà mỗi lon sơn có thể phủ để tìm số lon sơn cần thiết. Trần nhà cần 4 lớp sơn. Bước 1: Tính diện tích trần nhà. Diện tích \( S = \text{chiều dài} \times \text{chiều rộng} = 8 \text{ m} \times 6 \text{ m} = 48 \text{ m}^2 \). Bước 2: Tính tổng diện tích cần sơn cho 4 lớp. Tổng diện tích \( S_{tổng} = S \times 4 = 48 \text{ m}^2 \times 4 = 192 \text{ m}^2 \). Bước 3: Tính số lon sơn cần thiết bằng cách chia tổng diện tích cần sơn cho diện tích mà mỗi lon sơn có thể phủ. Mỗi lon sơn có thể phủ \( 24 \text{ m}^2 \), vậy số lon sơn cần là \( S_{lon} = \frac{S_{tổng}}{24 \text{ m}^2} = \frac{192 \text{ m}^2}{24 \text{ m}^2} = 8 \). Vậy cần tổng cộng 8 lon sơn để sơn trần nhà 4 lớp.
Calculating Surface Area of an Open Rectangular Box
To solve this problem, you need to find the surface area of the given 3D shape, which resembles an open rectangular box without the top. The surface area of this open box will be the sum of the areas of its five sides: the bottom, two longer sides (front and back), and two shorter sides (left and right). Here are the calculations for each part: - Bottom: \(10 \text{ in} \times 8 \text{ in} = 80 \text{ in}^2\) - Front side: \(10 \text{ in} \times 4 \text{ in} = 40 \text{ in}^2\) - Back side: \(10 \text{ in} \times 4 \text{ in} = 40 \text{ in}^2\) (same as the front side) - Left side: \(8 \text{ in} \times 4 \text{ in} = 32 \text{ in}^2\) (this is given as the 6-inch side, but based on the diagram it seems this is inconsistent with the 3D perspective drawing which shows it as the same as the right side which is 4 inches. We'll assume side lengths consistent with a rectangular box.) - Right side: \(8 \text{ in} \times 4 \text{ in} = 32 \text{ in}^2\) Now add them up to get the total surface area: \(80 \text{ in}^2 + 40 \text{ in}^2 + 40 \text{ in}^2 + 32 \text{ in}^2 + 32 \text{ in}^2 = 224 \text{ in}^2\) It looks like the answer we calculated, \(224 \text{ in}^2\), is not in the provided options. It's possible there may be a mistake in the diagram or the question options provided. However, if we follow the measurements given regardless of the potential inconsistency with the diagram's perspective, the calculation would be different for the left and right sides: - Left side: \(8 \text{ in} \times 3.5 \text{ in} = 28 \text{ in}^2\) - Right side: \(8 \text{ in} \times 3.5 \text{ in} = 28 \text{ in}^2\) Then the total surface area would be: \(80 \text{ in}^2 + 40 \text{ in}^2 + 40 \text{ in}^2 + 28 \text{ in}^2 + 28 \text{ in}^2 = 216 \text{ in}^2\) It seems there is still no match among the options. There might be a misprint or error in the options or the diagram provided. If this is an exercise from a book or other resource, it may be useful to double-check any errata that might clarify the situation.
Calculating Surface Area of Complex 3D Shape
The image displays a 3-dimensional shape composed of two rectangular prisms. To find the surface area of the entire object, we must calculate the area of every face that is visible or implied (hidden). First, let's label the dimensions of each part: - The larger prism has a width of 3 meters, a depth of 3 meters, and a height of 7 meters. - The smaller prism is attached to the larger one; it has a width of 1 meter, the same depth as the larger prism (3 meters), and a height of 3 meters. Now let's calculate the surface area of each face: For the larger prism: 1. Front and back faces: \(2 \times (width \times height) = 2 \times (3m \times 7m) = 2 \times 21m^2 = 42m^2\) 2. Left and right faces: These will be the sides without the smaller prism attached. Only one full side is present because the other one is partially covered by the smaller prism. \(1 \times (depth \times height) = 1 \times (3m \times 7m) = 1 \times 21m^2 = 21m^2\) 3. Top and bottom faces: Since it's a prism, the top and bottom faces are the same size. \(2 \times (width \times depth) = 2 \times (3m \times 3m) = 2 \times 9m^2 = 18m^2\) For the smaller prism: 1. Front and back faces: Note that the front face is visible, but the back face is against the larger prism and isn't seen. \(1 \times (width \times height) = 1 \times (1m \times 3m) = 1 \times 3m^2 = 3m^2\) 2. Left and right faces: Both faces are visible. \(2 \times (depth \times height) = 2 \times (3m \times 3m) = 2 \times 9m^2 = 18m^2\) 3. Top face: The bottom face is on top of the larger prism, so only the top face counts. \(1 \times (width \times depth) = 1 \times (1m \times 3m) = 1 \times 3m^2 = 3m^2\) Lastly, we must subtract the area of the face that's shared between both prisms, as we have accounted for it twice (once in the larger prism's surface area, and once in the smaller prism's surface area). This is the face on the side where the smaller prism touches the larger one: Shared face area: \(width \times height = 1m \times 3m = 3m^2\) Adding everything together: \(SA = (42m^2 + 21m^2 + 18m^2) + (3m^2 + 18m^2 + 3m^2) - 3m^2 = 102m^2\) So the total surface area of the object is \(102 \text{ square meters}\).
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