To compare the volumes of the sphere, cylinder, and cone, we first need to know their volume formulas.
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Volume of the sphere (V_s): \[ V_s = \frac{4}{3} \pi r^3 \]
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Volume of the cylinder (V_c): \[ V_c = \pi r^2 h \]
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Volume of the cone (V_cone): \[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Here, \(r\) is the radius and \(h\) is the height. Given that the cylinder and cone share the same height, we will use the same \(h\) for both.
Now, let's evaluate the provided relationships one by one.
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sphere = cylinder + cone: \[ V_s \neq V_c + V_{cone} \] Because \(V_c = \pi r^2 h\) and \(V_{cone} = \frac{1}{3} \pi r^2 h\), therefore:
\[ V_c + V_{cone} = \pi r^2 h + \frac{1}{3} \pi r^2 h = \frac{4}{3} \pi r^2 h \] So this statement is false. -
3 cones = 1 cylinder: \[ 3 \cdot V_{cone} = V_c \] Substituting the volumes gives: \[ 3 \cdot \frac{1}{3} \pi r^2 h = \pi r^2 h \] Thus, it is correct that \(3 \text{ cones} = 1 \text{ cylinder}\). So this statement is true.
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1 cylinder = 2 cones + 1 sphere: To check: \[ V_c = 2 V_{cone} + V_s \] Substituting gives: \[ \pi r^2 h \neq 2 \cdot \frac{1}{3} \pi r^2 h + \frac{4}{3} \pi r^3 \] This is a complex evaluation and doesn't hold true in general. Thus, this statement is false.
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sphere - cone = cylinder: To check: \[ V_s - V_{cone} = V_c \] Substituting gives: \[ \frac{4}{3} \pi r^3 - \frac{1}{3} \pi r^2 h \neq \pi r^2 h \] This would not generally hold true, hence this statement is false.
Thus, the only true statement among the options is:
\[ \text{3 cones} = \text{1 cylinder} \]
2 answers