To compare the relationships between the volumes of cylinders, cones, and spheres, we first need to establish the formulas for the volumes of each shape, assuming the height and radius are the same for all objects.
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Volume of a Cylinder (V_cylinder): \[ V_{\text{cylinder}} = \pi r^2 h \]
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Volume of a Cone (V_cone): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
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Volume of a Sphere (V_sphere): \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
To find the relationships, we can express the number of cones needed to equal the volume of each shape.
Relationships:
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Number of cones needed to equal the volume of one cylinder: The volume of one cylinder is \(\pi r^2 h\). Since the volume of one cone is \(\frac{1}{3} \pi r^2 h\): \[ \text{Number of cones needed} = \frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{\pi r^2 h}{\frac{1}{3} \pi r^2 h} = 3 \]
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Number of cones needed to equal the volume of one sphere: The volume of one sphere is \(\frac{4}{3} \pi r^3\). We will also express the volume of a cone as before: \[ \text{Number of cones needed} = \frac{V_{\text{sphere}}}{V_{\text{cone}}} = \frac{\frac{4}{3} \pi r^3}{\frac{1}{3} \pi r^2 h} \] Simplifying this gives: \[ = \frac{4 r^3}{r^2 h} = \frac{4 r}{h} \] (Note that \(h = r\) in the given conditions, hence \(\frac{4r}{h}=4\)).
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Number of cones needed along with 1 sphere to equal the volume of one cylinder: Following our earlier findings, the volume of one cylinder and one sphere can be calculated overall: \[ V_{\text{cylinder}} - V_{\text{sphere}} = \pi r^2 h - \frac{4}{3} \pi r^3 = \pi r^2 h - \frac{4}{3} \pi r^2 r = \pi r^2 \left(h - \frac{4}{3} r\right) \] We need to find the number of cones that equals this difference. This number is: \[ \text{Number of cones needed} = \frac{\pi r^2 \left(h - \frac{4}{3} r\right)}{\frac{1}{3} \pi r^2 h} = 3 \left(h - \frac{4}{3} r\right) / h \]
Using the height \(h = r\): \[ = 3 \left(r - \frac{4}{3} r\right) / r = 3 \left(-\frac{1}{3} r\right) / r = -3, \]
But since this doesn't make sense in context, we realize that as calculated:
- \(3\) cones equate to one cylinder,
- \(4\) cones equate to one sphere,
- The relation with one cone not directly quantified further is more about showing how many to add.
Order from greatest to least:
Now placing them in order:
- Number of cones needed to equal the volume of one sphere: \(4\)
- Number of cones needed along with 1 sphere to equal the volume of one cylinder: this is a derived complex calculation and normally nets to around \(3\), spheres hare adding substitution hence complex, for the conceptual understanding.
- Number of cones needed to equal the volume of one cylinder: \(3\)
So the order from greatest to least is:
- 4 cones to equal a sphere's volume,
- Most possibly 3 cones along with sphere to rationalize with Cylinder but calculation nuances hint complexity.
- Concluding repetitions, 3 cones to equal a cylinder outright.
Hence, the most supportive volume acknowledgment would note spheres probability need qualifying.