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1. The figure shows a solid which consists of a hemisphere and a right circular cylinder with the same base. The height of the circular cylinder is twice its base radius. If the volume of the solid is \( \frac{512 \pi}{3} \mathrm{~cm}^{3} \), find the height of the circular cylinder. 2. In the figure, the radius of the sphere and the base radius of the right circular cylinder are the same. The base radius and the height of the circular cylinder are the same. If the total volume of the two solids is \( 63 \pi \mathrm{~cm}^{3} \), find the diameter of the sphere.

Ask by Warren Bob. in Hong Kong
Mar 16,2025

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### 問題 1 已知一個固體由一個半球和一個圓柱組成,兩者的底面半徑相同。圓柱的高度是其底面半徑的兩倍。固體的體積為 \( \frac{512 \pi}{3} \mathrm{~cm}^{3} \)。 **答案:** 圓柱的高度為 \( 8 \, \text{cm} \)。 --- ### 問題 2 已知一個球體和一個圓柱的底面半徑相同,圓柱的底面半徑和高度相同。兩個固體的總體積為 \( 63 \pi \mathrm{~cm}^{3} \)。 **答案:** 球的直徑為 \( 6 \, \text{cm} \)。

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### 問題 1 已知一個固體由一個半球和一個圓柱組成,兩者的底面半徑相同。圓柱的高度是其底面半徑的兩倍。固體的體積為 \( \frac{512 \pi}{3} \mathrm{~cm}^{3} \)。 **已知條件:** - 半球的體積公式:\( V_{h} = \frac{2}{3} \pi r^3 \) - 圓柱的體積公式:\( V_{c} = \pi r^2 h \) - 圓柱的高度 \( h = 2r \) - 總體積 \( V = V_{h} + V_{c} = \frac{512 \pi}{3} \) **步驟:** 1. 將圓柱的高度代入體積公式: \[ V_{c} = \pi r^2 (2r) = 2 \pi r^3 \] 2. 總體積公式變為: \[ V = V_{h} + V_{c} = \frac{2}{3} \pi r^3 + 2 \pi r^3 \] 3. 將 \( 2 \pi r^3 \) 轉換為相同的分母: \[ 2 \pi r^3 = \frac{6}{3} \pi r^3 \] 4. 總體積公式變為: \[ V = \frac{2}{3} \pi r^3 + \frac{6}{3} \pi r^3 = \frac{8}{3} \pi r^3 \] 5. 設定等式: \[ \frac{8}{3} \pi r^3 = \frac{512 \pi}{3} \] 6. 兩邊同除以 \( \frac{\pi}{3} \): \[ 8r^3 = 512 \] 7. 兩邊同除以 8: \[ r^3 = 64 \] 8. 開立方根: \[ r = 4 \, \text{cm} \] 9. 計算圓柱的高度: \[ h = 2r = 2 \times 4 = 8 \, \text{cm} \] **答案:** 圓柱的高度為 \( 8 \, \text{cm} \)。 --- ### 問題 2 已知一個球體和一個圓柱的底面半徑相同,圓柱的底面半徑和高度相同。兩個固體的總體積為 \( 63 \pi \mathrm{~cm}^{3} \)。 **已知條件:** - 球體的體積公式:\( V_{s} = \frac{4}{3} \pi r^3 \) - 圓柱的體積公式:\( V_{c} = \pi r^2 h \) - 圓柱的高度 \( h = r \) - 總體積 \( V = V_{s} + V_{c} = 63 \pi \) **步驟:** 1. 將圓柱的高度代入體積公式: \[ V_{c} = \pi r^2 r = \pi r^3 \] 2. 總體積公式變為: \[ V = V_{s} + V_{c} = \frac{4}{3} \pi r^3 + \pi r^3 \] 3. 將 \( \pi r^3 \) 轉換為相同的分母: \[ \pi r^3 = \frac{3}{3} \pi r^3 \] 4. 總體積公式變為: \[ V = \frac{4}{3} \pi r^3 + \frac{3}{3} \pi r^3 = \frac{7}{3} \pi r^3 \] 5. 設定等式: \[ \frac{7}{3} \pi r^3 = 63 \pi \] 6. 兩邊同除以 \( \pi \): \[ \frac{7}{3} r^3 = 63 \] 7. 兩邊同乘以 3: \[ 7r^3 = 189 \] 8. 兩邊同除以 7: \[ r^3 = 27 \] 9. 開立方根: \[ r = 3 \, \text{cm} \] 10. 計算球的直徑: \[ \text{直徑} = 2r = 2 \times 3 = 6 \, \text{cm} \] **答案:** 球的直徑為 \( 6 \, \text{cm} \)。

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To solve the first problem, we recognize that the volume of the solid consists of the volume of the hemisphere and the cylinder. The volume \( V \) of a hemisphere is given by \( \frac{2}{3} \pi r^3 \), and the volume of a cylinder is \( \pi r^2 h \). Given that the height \( h \) of the cylinder is twice the base radius \( r \), we can substitute \( h = 2r \) into the volume formula. So, the total volume becomes: \[ V = \frac{2}{3} \pi r^3 + \pi r^2 (2r) = \frac{2}{3} \pi r^3 + 2\pi r^3 = \frac{2}{3} \pi r^3 + \frac{6}{3} \pi r^3 = \frac{8}{3} \pi r^3 \] We set this equal to the given volume \( \frac{512 \pi}{3} \): \[ \frac{8}{3} \pi r^3 = \frac{512 \pi}{3} \] Solving for \( r^3 \), we find \( 8r^3 = 512 \), which gives \( r^3 = 64 \), leading to \( r = 4 \) cm. The height of the cylinder is: \[ h = 2r = 2 \times 4 = 8 \text{ cm.} \] For the second problem, we have a sphere and a cylinder with the same radius \( r \) and the cylinder's height \( h = r \). The volume of the sphere is \( \frac{4}{3} \pi r^3 \), and the volume of the cylinder is \( \pi r^2 h = \pi r^3 \). Thus, the total volume is: \[ V = \frac{4}{3} \pi r^3 + \pi r^3 = \frac{4}{3} \pi r^3 + \frac{3}{3} \pi r^3 = \frac{7}{3} \pi r^3 \] Setting this equal to \( 63 \pi \): \[ \frac{7}{3} \pi r^3 = 63 \pi \] Dividing both sides by \( \pi \) and multiplying by \( \frac{3}{7} \): \[ r^3 = 63 \times \frac{3}{7} = 27 \] Thus, \( r = 3 \) cm. The diameter of the sphere is \( 2r = 2 \times 3 = 6 \) cm.

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