To fill in the blanks regarding the relationships among the volumes of cylinders, cones, and spheres, we need to recall the formulas for each shape's volume, assuming the height and radius are the same.
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Volume of a Cylinder (V_cylinder): \[ V_{cylinder} = \pi r^2 h \]
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Volume of a Cone (V_cone): \[ V_{cone} = \frac{1}{3} \pi r^2 h \] This shows that a cone's volume is one-third that of a cylinder with the same radius and height.
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Volume of a Sphere (V_sphere): \[ V_{sphere} = \frac{4}{3} \pi r^3 \] When comparing the sphere to the cylinder or cone, we'll need to relate it to volume forms.
Using the formula relationships and provided options:
A. One sphere + ___ cone(s) = one cylinder
We know that the volume of one sphere is \( \frac{4}{3} \pi r^3 \) and that of a cylinder is \( \pi r^2 h \). In terms of cones, we can derive the necessary number of cones based on the cone's volume:
- One cone = \( \frac{1}{3} \) of a cylinder volume.
- Therefore, we need to find the number of cones that equate the volume of the sphere and one cylinder.
From previous knowledge: \[ \text{One sphere} \approx \frac{3}{4} \text{ of a cylinder} \implies 1 \text{ sphere} + 2 \text{ cones} = 1 \text{ cylinder} \] The answer is 2.
B. ___ cone(s) = one cylinder
We found earlier that one cone is \( \frac{1}{3} \) of a cylinder. Therefore: \[ \text{1 cylinder} = 3 \text{ cones} \] The answer is 3.
C. ___ cone(s) = one sphere
From our earlier calculations, we already had the relationship of volume where: \[ \text{One sphere} + \text{2 cones} = \text{One cylinder} \] If one sphere can be represented as being equivalent to \( \frac{3}{4} \) of a cylinder, and since one cone needs to be calculated against the sphere's volume using the same proportionality, we can see: \[ \text{1 sphere} = 3 \text{ cones} \] We would actually need that equal to \( 2 \text{ cones} \) based on that fraction needed from earlier: one cone effectively fills up a portion with multiple adjustments. The answer here is 3.
D. One sphere = ___ cylinder
From earlier: \[ \text{1 sphere} = \frac{3}{4} \text{ of a cylinder} \] So in terms of fraction: We get that: One sphere = \( \frac{2}{3} \) of a cylinder The answer is 2/3.
E. One cone = ___ cylinder
Once again, since we know: \[ \text{1 cone} = \frac{1}{3} \text{ of a cylinder} \] The answer is 1/3.
To Summarize:
A. 2
B. 3
C. 3
D. 2/3
E. 1/3
Thus, we have:
- A. 2
- B. 3
- C. 3
- D. 2/3
- E. 1/3