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Example Question - volume formula

Here are examples of questions we've helped users solve.

Calculating Volume by Disk Method for Intersection of Curves

Para calcular el volumen del sólido generado al girar la región acotada por las curvas \(y = x^2\) y \( y = x^3 \) alrededor del eje \(X\), utilizaremos el método del disco. Primero, debemos determinar los puntos donde las curvas se intersectan, igualando las ecuaciones: \[ x^2 = x^3 \] \[ x^2(1 - x) = 0 \] Esto nos da dos puntos de intersección: \(x = 0\) y \(x = 1\). Ahora, para aplicar el método del disco, integramos π por el radio al cuadrado (desde la curva más cercana al eje \(X\) hasta la más lejana) respecto a \(x\), entre \(x = 0\) y \(x = 1\). El radio es la distancia en y entre las dos curvas, o sea, \( x^2 - x^3 \). Así que el volumen \(V\) es: \[ V = \pi \int_{0}^{1} (x^2 - x^3)^2 dx \] \[ V = \pi \int_{0}^{1} (x^4 - 2x^5 + x^6) dx \] \[ V = \pi \left[ \frac{x^5}{5} - 2\frac{x^6}{6} + \frac{x^7}{7} \right]_{0}^{1} \] \[ V = \pi \left[ \frac{1}{5} - \frac{2}{6} + \frac{1}{7} \right] \] \[ V = \pi \left[ \frac{1}{5} - \frac{1}{3} + \frac{1}{7} \right] \] \[ V = \pi \left[ \frac{21}{105} - \frac{35}{105} + \frac{15}{105} \right] \] \[ V = \pi \left[ \frac{1}{105} \right] \] \[ V = \frac{\pi}{105} \] Por lo tanto, el volumen del sólido generado es \(\frac{\pi}{105}\) unidades cúbicas.

Calculating Volume of Rectangular Prisms and Converting to Liters

Pour résoudre cette question, il faut commencer par calculer le volume de chaque pavé droit en centimètres cubes (cm³) puis convertir le résultat en litres, sachant que 1 litre équivaut à 1 000 centimètres cubes. Pour le pavé droit a: Le volume \( V \) est donné par la formule \( V = longueur \times largeur \times hauteur \). Ici la longueur est 5 dm (ce qui équivaut à 50 cm), la largeur est 6 dm (60 cm) et la hauteur est 3 dm (30 cm). Donc, \( V_a = 50 \times 60 \times 30 \). Faisons le calcul: \( V_a = 50 \times 60 \times 30 \) \( V_a = 3000 \times 30 \) \( V_a = 90 000 \) cm³ Ensuite, convertissons les centimètres cubes en litres: \( 90 000 \) cm³ équivaut à \( 90 \) litres car \( 90 000 \div 1 000 = 90 \). Pour le pavé droit b: La longueur est 7 dm (70 cm), la largeur est 2,1 dm (21 cm) et la hauteur est 3 dm (30 cm). Donc, \( V_b = 70 \times 21 \times 30 \). Faisons le calcul: \( V_b = 70 \times 21 \times 30 \) \( V_b = 1470 \times 30 \) \( V_b = 44 100 \) cm³ Et convertissons les centimètres cubes en litres: \( 44 100 \) cm³ équivaut à \( 44,1 \) litres car \( 44 100 \div 1 000 = 44,1 \). Conclusion: Le pavé droit a a un volume de 90 litres d'eau, et le pavé droit b a un volume de 44,1 litres d'eau.

Geometry Problem: Calculating Height of a Rectangular Prism

Dựa trên hình ảnh bạn cung cấp, đây là một bài toán hình học liên quan đến hình hợp. Theo thông tin được đề cập, hình hợp này có chiều dài là 4,5dm, chiều rộng 2,6dm và thể tích của hình hợp là 17,4dm³. Bài toán yêu cầu tính chiều cao của hình hợp đó. Để tính chiều cao của hình hợp, chúng ta sử dụng công thức tính thể tích: Thể tích hình hợp = Chiều dài x Chiều rộng x Chiều cao Mặc định, thể tích của hình hợp đã được cung cấp trong đề bài. Ta cần tìm chiều cao, vì vậy ta sẽ biến đổi công thức để chiều cao là phần tử cần tìm: Chiều cao = Thể tích hình hợp / (Chiều dài x Chiều rộng) Thay số liệu cụ thể vào công thức, ta có: Chiều cao = 17,4dm³ / (4,5dm x 2,6dm) Chiều cao = 17,4 / 11,7 Chiều cao ≈ 1,49dm Vậy chiều cao của hình hợp là khoảng 1,49dm.

Finding Width of Right Rectangular Prism from Volume, Height, and Length

The problem is asking us to find out the width of a right rectangular prism given the volume, the height, and the length. We know that the volume (V) of a rectangular prism is given by the formula: V = length × width × height The volume (V) is given as 15,375.92 cubic yards, the height (h) is 8.19 yards, and the length (l) is 11.2 yards. We need to find the width (w). We can rearrange the formula to solve for width: width = V / (length × height) Now let's substitute the given values: width = 15,375.92 / (11.2 × 8.19) width = 15,375.92 / 91.728 width ≈ 167.6 yards So, the width of the right rectangular prism is approximately 167.6 yards.

Calculating the Volume of a Rectangular Prism

The image displays a rectangular prism (also known as a rectangular solid or a cuboid) with labeled dimensions. The dimensions given are 13 cm for the length, 5 cm for the width (or depth), and 6 cm for the height. If the question from the image is to find the volume of the rectangular prism, the formula to use is: Volume = length × width × height Applying the given dimensions to this formula: Volume = 13 cm × 5 cm × 6 cm Volume = 65 cm² × 6 cm Volume = 390 cm³ The volume of the rectangular prism is 390 cubic centimeters.

Calculating the Volume of a Rectangular Prism

The image depicts a rectangular prism, where the dimensions are given as follows: - Length (l): \( 16\frac{1}{2} \) cm or 16.5 cm - Width (w): 4 cm - Height (h): \( 9\frac{3}{4} \) cm or 9.75 cm Assuming you are looking to calculate the volume of this rectangular prism, the formula to use is: \[ \text{Volume} = l \times w \times h \] Let's insert the given values: \[ \text{Volume} = 16.5 \text{ cm} \times 4 \text{ cm} \times 9.75 \text{ cm} \] Now calculate each multiplication step by step: \[ \text{Volume} = 66 \text{ cm}^2 \times 9.75 \text{ cm} \] \[ \text{Volume} = 643.5 \text{ cm}^3 \] So, the volume of the given rectangular prism is 643.5 cubic centimeters.

Calculating Volume of a Rectangular Prism

The volume of a rectangular prism (which is the shape shown in the image) is calculated by multiplying its length, width, and height. From the image, the length is given as 9 mm, the width as 8 mm, and the height as 2 mm. Volume = length × width × height Volume = 9 mm × 8 mm × 2 mm Volume = 72 mm² × 2 mm Volume = 144 mm³ So the volume of the prism is 144 cubic millimeters.

Calculating the Volume of a Composite Solid

To find the volume of this composite solid, we need to break it into smaller parts whose volumes we can calculate easily and then sum up those volumes. By looking at the shapes, we can split the figure into two rectangular prisms. Let's call the one on the left "Prism A" and the one on the right "Prism B". 1. **Volume of Prism A**: For Prism A, we have the dimensions 2 cm by 6 cm by 10 cm. The volume (V) of a rectangular prism can be found using the formula V = length × width × height. Therefore, V of Prism A = 2 cm × 6 cm × 10 cm = 120 cm³. 2. **Volume of Prism B**: For Prism B, the dimensions are not as straightforward, so we need to calculate the missing dimensions first. We can see that the height of the prism is 5 cm. The length of the prism B can be determined by subtracting the length of prism A from the total length. That is 10 cm (total length) - 6 cm (length of A) = 4 cm. So, the length of prism B is 4 cm. The width of prism B is the total width, which is 8 cm. Therefore, V of Prism B = 5 cm × 4 cm × 8 cm = 160 cm³. Now, we add the volumes of the two prisms together: Total Volume = Volume of Prism A + Volume of Prism B Total Volume = 120 cm³ + 160 cm³ = 280 cm³. The volume of the composite solid is 280 cm³.

Calculating Volumes of Rectangular Prisms and Polyhedron

To calculate the volume of each rectangular prism and the combined polyhedron, use the formula: Volume of a rectangular prism = length × width × height For Rectangular Prism 1: - Length = 11 mm - Width = 5 mm - Height = 8 mm Volume = 11 mm × 5 mm × 8 mm = 440 cubic millimeters For Rectangular Prism 2: - Length = 15 mm - Width = 8 mm - Height = 8 mm Volume = 15 mm × 8 mm × 8 mm = 960 cubic millimeters Now for the combined polyhedron which is the sum of the volumes of the two prisms: Volume = Volume of Prism 1 + Volume of Prism 2 Volume = 440 mm³ + 960 mm³ = 1400 mm³ So, the answers are: A) The volume of Rectangular Prism 1 is 440 cubic millimeters. B) The volume of Rectangular Prism 2 is 960 cubic millimeters. C) The volume of the polyhedron is 1400 cubic millimeters.