Alege rezolvarea exercițiului:
Exercițiul 1 (gratuit)
Rezolvare scurtă
Rândul 1
\( R = 8\text{ dm}, G = 10\text{ dm} \) \( \mathcal{A}_l = 2\pi RG = 2\pi \cdot 8 \cdot 10 = 160\pi\text{ dm}^2 \) \( \mathcal{A}_b = \pi R^2 = \pi \cdot 8^2 = 64\pi\text{ dm}^2 \) \( \mathcal{A}_t = \mathcal{A}_l + 2\mathcal{A}_b = 160\pi + 2 \cdot 64\pi = 288\pi\text{ dm}^2 \) \( \mathcal{V} = \pi R^2 G = \pi \cdot 8^2 \cdot 10 = 640\pi\text{ dm}^3 \)Rândul 2
\( G = 8\text{ m}, \mathcal{V} = 288\pi\text{ m}^3 \) \( \mathcal{V} = \pi R^2 G \Rightarrow 288\pi = \pi R^2 \cdot 8 \) \( R^2 = \frac{\cancel{288\pi}^{36}}{\cancel{8\pi}_1} = 36 \Rightarrow R = 6\text{ m} \) \( \mathcal{A}_l = 2\pi RG = 2\pi \cdot 6 \cdot 8 = 96\pi\text{ m}^2 \) \( \mathcal{A}_b = \pi R^2 = \pi \cdot 6^2 = 36\pi\text{ m}^2 \) \( \mathcal{A}_t = \mathcal{A}_l + 2\mathcal{A}_b = 96\pi + 2 \cdot 36\pi = 168\pi\text{ m}^2 \)Rândul 3
\( \mathcal{A}_l = 60\pi\text{ m}^2, \mathcal{A}_t = 110\pi\text{ m}^2 \) \( \mathcal{A}_t = \mathcal{A}_l + 2\mathcal{A}_b \Rightarrow 110\pi = 60\pi + 2\mathcal{A}_b \) \( 2\mathcal{A}_b = 50\pi \Rightarrow \mathcal{A}_b = 25\pi\text{ m}^2 \) \( \mathcal{A}_b = \pi R^2 \Rightarrow 25\pi = \pi R^2 \Rightarrow R^2 = 25 \Rightarrow R = 5\text{ m} \) \( \mathcal{A}_l = 2\pi RG \Rightarrow 60\pi = 2\pi \cdot 5 \cdot G \) \( G = \frac{\cancel{60\pi}^{6}}{\cancel{10\pi}_1} = 6\text{ m} \) \( \mathcal{V} = \pi R^2 G = \pi \cdot 5^2 \cdot 6 = 150\pi\text{ m}^3 \)Rândul 4
\( R = 3\text{ cm}, \mathcal{V} = 90\pi\text{ cm}^3 \) \( \mathcal{V} = \pi R^2 G \Rightarrow 90\pi = \pi \cdot 3^2 \cdot G \) \( G = \frac{\cancel{90\pi}^{10}}{\cancel{9\pi}_1} = 10\text{ cm} \) \( \mathcal{A}_l = 2\pi RG = 2\pi \cdot 3 \cdot 10 = 60\pi\text{ cm}^2 \) \( \mathcal{A}_b = \pi R^2 = \pi \cdot 3^2 = 9\pi\text{ cm}^2 \) \( \mathcal{A}_t = \mathcal{A}_l + 2\mathcal{A}_b = 60\pi + 2 \cdot 9\pi = 78\pi\text{ cm}^2 \)Rândul 5
\( R = 12\text{ m}, \mathcal{A}_l = 144\pi\text{ m}^2 \) \( \mathcal{A}_l = 2\pi RG \Rightarrow 144\pi = 2\pi \cdot 12 \cdot G \) \( G = \frac{\cancel{144\pi}^{6}}{\cancel{24\pi}_1} = 6\text{ m} \) \( \mathcal{A}_b = \pi R^2 = \pi \cdot 12^2 = 144\pi\text{ m}^2 \) \( \mathcal{A}_t = \mathcal{A}_l + 2\mathcal{A}_b = 144\pi + 2 \cdot 144\pi = 432\pi\text{ m}^2 \) \( \mathcal{V} = \pi R^2 G = \pi \cdot 12^2 \cdot 6 = 864\pi\text{ m}^3 \)Tabelul completat
\[ \begin{array}{|c|c|c|c|c|} \hline R & G & \mathcal{A}_l & \mathcal{A}_t & \mathcal{V} \\ \hline 8\text{ dm} & 10\text{ dm} & 160\pi\text{ dm}^2 & 288\pi\text{ dm}^2 & 640\pi\text{ dm}^3 \\ \hline 6\text{ m} & 8\text{ m} & 96\pi\text{ m}^2 & 168\pi\text{ m}^2 & 288\pi\text{ m}^3 \\ \hline 5\text{ m} & 6\text{ m} & 60\pi\text{ m}^2 & 110\pi\text{ m}^2 & 150\pi\text{ m}^3 \\ \hline 3\text{ cm} & 10\text{ cm} & 60\pi\text{ cm}^2 & 78\pi\text{ cm}^2 & 90\pi\text{ cm}^3 \\ \hline 12\text{ m} & 6\text{ m} & 144\pi\text{ m}^2 & 432\pi\text{ m}^2 & 864\pi\text{ m}^3 \\ \hline \end{array} \]Rezolvare detaliată
Pentru a completa datele din tabel referitoare la cilindrul circular drept, vom folosi
următoarele formule specifice:
- Aria bazei: \( \mathcal{A}_b = \pi R^2 \)
- Aria laterală: \( \mathcal{A}_l = 2\pi RG \)
- Aria totală: \( \mathcal{A}_t = \mathcal{A}_l + 2\mathcal{A}_b = 2\pi R(G + R) \)
- Volumul: \( \mathcal{V} = \pi R^2 G \) (deoarece generatorul \( G \) este egal cu înălțimea \( h \))
Pasul 1: Rezolvarea pentru rândul 1
Se cunosc: \( R = 8\text{ dm} \), \( G = 10\text{ dm} \). - Calculăm aria laterală \( \mathcal{A}_l \): \[ \mathcal{A}_l = 2\pi RG = 2\pi \cdot 8 \cdot 10 = 160\pi\text{ dm}^2 \] - Calculăm aria bazei \( \mathcal{A}_b \): \[ \mathcal{A}_b = \pi R^2 = \pi \cdot 8^2 = 64\pi\text{ dm}^2 \] - Calculăm aria totală \( \mathcal{A}_t \): \[ \mathcal{A}_t = 160\pi + 2 \cdot 64\pi = 160\pi + 128\pi \] \[ \mathcal{A}_t = 288\pi\text{ dm}^2 \] - Calculăm volumul \( \mathcal{V} \): \[ \mathcal{V} = \pi R^2 G = \pi \cdot 8^2 \cdot 10 = 64\pi \cdot 10 = 640\pi\text{ dm}^3 \]Pasul 2: Rezolvarea pentru rândul 2
Se cunosc: \( G = 8\text{ m} \), \( \mathcal{V} = 288\pi\text{ m}^3 \). - Determinăm raza bazei \( R \): \[ \mathcal{V} = \pi R^2 G \Rightarrow 288\pi = \pi R^2 \cdot 8 \] \[ R^2 = \frac{\cancel{288\pi}^{36}}{\cancel{8\pi}_1} = 36 \Rightarrow R = 6\text{ m} \] - Calculăm aria laterală \( \mathcal{A}_l \): \[ \mathcal{A}_l = 2\pi RG = 2\pi \cdot 6 \cdot 8 = 96\pi\text{ m}^2 \] - Calculăm aria bazei \( \mathcal{A}_b \): \[ \mathcal{A}_b = \pi R^2 = \pi \cdot 6^2 = 36\pi\text{ m}^2 \] - Calculăm aria totală \( \mathcal{A}_t \): \[ \mathcal{A}_t = 96\pi + 2 \cdot 36\pi = 96\pi + 72\pi = 168\pi\text{ m}^2 \]Pasul 3: Rezolvarea pentru rândul 3
Se cunosc: \( \mathcal{A}_l = 60\pi\text{ m}^2 \), \( \mathcal{A}_t = 110\pi\text{ m}^2 \). - Determinăm aria bazei \( \mathcal{A}_b \): \[ \mathcal{A}_t = \mathcal{A}_l + 2\mathcal{A}_b \Rightarrow 110\pi = 60\pi + 2\mathcal{A}_b \] \[ 2\mathcal{A}_b = 110\pi - 60\pi = 50\pi \Rightarrow \mathcal{A}_b = 25\pi\text{ m}^2 \] - Determinăm raza bazei \( R \): \[ \mathcal{A}_b = \pi R^2 \Rightarrow 25\pi = \pi R^2 \Rightarrow R^2 = 25 \] \[ R = \sqrt{25} = 5\text{ m} \] - Determinăm generatoarea \( G \): \[ \mathcal{A}_l = 2\pi RG \Rightarrow 60\pi = 2\pi \cdot 5 \cdot G \] \[ G = \frac{\cancel{60\pi}^{6}}{\cancel{10\pi}_1} = 6\text{ m} \] - Calculăm volumul \( \mathcal{V} \): \[ \mathcal{V} = \pi R^2 G = \pi \cdot 5^2 \cdot 6 = 25\pi \cdot 6 = 150\pi\text{ m}^3 \]Pasul 4: Rezolvarea pentru rândul 4
Se cunosc: \( R = 3\text{ cm} \), \( \mathcal{V} = 90\pi\text{ cm}^3 \). - Determinăm generatoarea \( G \): \[ \mathcal{V} = \pi R^2 G \Rightarrow 90\pi = \pi \cdot 3^2 \cdot G \] \[ G = \frac{\cancel{90\pi}^{10}}{\cancel{9\pi}_1} = 10\text{ cm} \] - Calculăm aria laterală \( \mathcal{A}_l \): \[ \mathcal{A}_l = 2\pi RG = 2\pi \cdot 3 \cdot 10 = 60\pi\text{ cm}^2 \] - Calculăm aria bazei \( \mathcal{A}_b \): \[ \mathcal{A}_b = \pi R^2 = \pi \cdot 3^2 = 9\pi\text{ cm}^2 \] - Calculăm aria totală \( \mathcal{A}_t \): \[ \mathcal{A}_t = 60\pi + 2 \cdot 9\pi = 60\pi + 18\pi \] \[ \mathcal{A}_t = 78\pi\text{ cm}^2 \]Pasul 5: Rezolvarea pentru rândul 5
Se cunosc: \( R = 12\text{ m} \), \( \mathcal{A}_l = 144\pi\text{ m}^2 \). - Determinăm generatoarea \( G \): \[ \mathcal{A}_l = 2\pi RG \Rightarrow 144\pi = 2\pi \cdot 12 \cdot G \] \[ G = \frac{\cancel{144\pi}^{6}}{\cancel{24\pi}_1} = 6\text{ m} \] - Calculăm aria bazei \( \mathcal{A}_b \): \[ \mathcal{A}_b = \pi R^2 = \pi \cdot 12^2 = 144\pi\text{ m}^2 \] - Calculăm aria totală \( \mathcal{A}_t \): \[ \mathcal{A}_t = 144\pi + 2 \cdot 144\pi = 144\pi + 288\pi \] \[ \mathcal{A}_t = 432\pi\text{ m}^2 \] - Calculăm volumul \( \mathcal{V} \): \[ \mathcal{V} = \pi R^2 G = \pi \cdot 12^2 \cdot 6 = 144\pi \cdot 6 = 864\pi\text{ m}^3 \]Pasul 6: Centralizarea rezultatelor
Completăm tabelul cu valorile obținute în urma calculelor: \[ \begin{array}{|c|c|c|c|c|} \hline R & G & \mathcal{A}_l & \mathcal{A}_t & \mathcal{V} \\ \hline 8 \text{ dm} & 10 \text{ dm} & 160\pi \text{ dm}^2 & 288\pi \text{ dm}^2 & 640\pi \text{ dm}^3 \\ \hline 6 \text{ m} & 8 \text{ m} & 96\pi \text{ m}^2 & 168\pi \text{ m}^2 & 288\pi \text{ m}^3 \\ \hline 5 \text{ m} & 6 \text{ m} & 60\pi \text{ m}^2 & 110\pi \text{ m}^2 & 150\pi \text{ m}^3 \\ \hline 3 \text{ cm} & 10 \text{ cm} & 60\pi \text{ cm}^2 & 78\pi \text{ cm}^2 & 90\pi \text{ cm}^3 \\ \hline 12 \text{ m} & 6 \text{ m} & 144\pi \text{ m}^2 & 432\pi \text{ m}^2 & 864\pi \text{ m}^3 \\ \hline \end{array} \]Rezolvare scurtă:
Row 1: \( \mathcal{A}_l = 2\pi \cdot 8 \cdot 10 = 160\pi\text{ dm}^2 \),
\( \mathcal{A}_t = 160\pi + 2\pi \cdot 8^2 = 288\pi\text{ dm}^2 \),
\( \mathcal{V} = \pi \cdot 8^2 \cdot 10 = 640\pi\text{ dm}^3 \).
Row 2: \( R = \sqrt{\frac{288\pi}{8\pi}} = 6\text{ m} \),
\( \mathcal{A}_l = 2\pi \cdot 6 \cdot 8 = 96\pi\text{ m}^2 \),
\( \mathcal{A}_t = 96\pi + 2\pi \cdot 6^2 = 168\pi\text{ m}^2 \).
Row 3: \( \mathcal{A}_b = \frac{110\pi - 60\pi}{2} = 25\pi\text{ m}^2 \Rightarrow R = 5\text{ m} \),
\( G = \frac{60\pi}{2\pi \cdot 5} = 6\text{ m} \),
\( \mathcal{V} = \pi \cdot 5^2 \cdot 6 = 150\pi\text{ m}^3 \).
Row 4: \( G = \frac{90\pi}{\pi \cdot 3^2} = 10\text{ cm} \),
\( \mathcal{A}_l = 2\pi \cdot 3 \cdot 10 = 60\pi\text{ cm}^2 \),
\( \mathcal{A}_t = 60\pi + 2\pi \cdot 3^2 = 78\pi\text{ cm}^2 \).
Row 5: \( G = \frac{144\pi}{2\pi \cdot 12} = 6\text{ m} \),
\( \mathcal{A}_t = 144\pi + 2\pi \cdot 12^2 = 432\pi\text{ m}^2 \),
\( \mathcal{V} = \pi \cdot 12^2 \cdot 6 = 864\pi\text{ m}^3 \).
Tabel completat:
\[
\begin{array}{|c|c|c|c|c|}
\hline
R & G & \mathcal{A}_l & \mathcal{A}_t & \mathcal{V} \\ \hline
8\text{ dm} & 10\text{ dm} & 160\pi\text{ dm}^2 & 288\pi\text{ dm}^2 & 640\pi\text{ dm}^3 \\ \hline
6\text{ m} & 8\text{ m} & 96\pi\text{ m}^2 & 168\pi\text{ m}^2 & 288\pi\text{ m}^3 \\ \hline
5\text{ m} & 6\text{ m} & 60\pi\text{ m}^2 & 110\pi\text{ m}^2 & 150\pi\text{ m}^3 \\ \hline
3\text{ cm} & 10\text{ cm} & 60\pi\text{ cm}^2 & 78\pi\text{ cm}^2 & 90\pi\text{ cm}^3 \\ \hline
12\text{ m} & 6\text{ m} & 144\pi\text{ m}^2 & 432\pi\text{ m}^2 & 864\pi\text{ m}^3 \\ \hline
\end{array}
\]
