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Math  /  Geometry

Question5. Marie-Claude has a series of four nested funnels in her kitchen that are similar to the one shown in the diagram. If the other three funnels have top diameters of 10 cm,8 cm10 \mathrm{~cm}, 8 \mathrm{~cm}, and 6 cm , find the measures of the remaining parts for all three funnels.

Studdy Solution

STEP 1

1. All funnels are similar, meaning they have the same shape but different sizes.
2. The dimensions of the given funnel are: top diameter = 12 cm, total height = 16 cm, height of the conical part = 10 cm, and diameter of the narrow tube = 2 cm.
3. The similarity ratio can be used to find corresponding dimensions of other funnels.

STEP 2

1. Determine the similarity ratio for each funnel.
2. Calculate the total height for each funnel.
3. Calculate the height of the conical part for each funnel.
4. Calculate the diameter of the narrow tube for each funnel.

STEP 3

Determine the similarity ratio for each funnel based on the top diameter. The given funnel has a top diameter of 12 cm. For each of the other funnels, the similarity ratio is given by:
Similarity Ratio=Top Diameter of Other FunnelTop Diameter of Given Funnel \text{Similarity Ratio} = \frac{\text{Top Diameter of Other Funnel}}{\text{Top Diameter of Given Funnel}}
For the 10 cm funnel: Similarity Ratio=1012=56 \text{Similarity Ratio} = \frac{10}{12} = \frac{5}{6}
For the 8 cm funnel: Similarity Ratio=812=23 \text{Similarity Ratio} = \frac{8}{12} = \frac{2}{3}
For the 6 cm funnel: Similarity Ratio=612=12 \text{Similarity Ratio} = \frac{6}{12} = \frac{1}{2}

STEP 4

Calculate the total height for each funnel using the similarity ratio. The total height of the given funnel is 16 cm.
For the 10 cm funnel: Total Height=16×56=806=13.33 cm \text{Total Height} = 16 \times \frac{5}{6} = \frac{80}{6} = 13.33 \, \text{cm}
For the 8 cm funnel: Total Height=16×23=323=10.67 cm \text{Total Height} = 16 \times \frac{2}{3} = \frac{32}{3} = 10.67 \, \text{cm}
For the 6 cm funnel: Total Height=16×12=8 cm \text{Total Height} = 16 \times \frac{1}{2} = 8 \, \text{cm}

STEP 5

Calculate the height of the conical part for each funnel using the similarity ratio. The height of the conical part of the given funnel is 10 cm.
For the 10 cm funnel: Height of Conical Part=10×56=506=8.33 cm \text{Height of Conical Part} = 10 \times \frac{5}{6} = \frac{50}{6} = 8.33 \, \text{cm}
For the 8 cm funnel: Height of Conical Part=10×23=203=6.67 cm \text{Height of Conical Part} = 10 \times \frac{2}{3} = \frac{20}{3} = 6.67 \, \text{cm}
For the 6 cm funnel: Height of Conical Part=10×12=5 cm \text{Height of Conical Part} = 10 \times \frac{1}{2} = 5 \, \text{cm}

STEP 6

Calculate the diameter of the narrow tube for each funnel using the similarity ratio. The diameter of the narrow tube of the given funnel is 2 cm.
For the 10 cm funnel: Diameter of Narrow Tube=2×56=106=1.67 cm \text{Diameter of Narrow Tube} = 2 \times \frac{5}{6} = \frac{10}{6} = 1.67 \, \text{cm}
For the 8 cm funnel: Diameter of Narrow Tube=2×23=43=1.33 cm \text{Diameter of Narrow Tube} = 2 \times \frac{2}{3} = \frac{4}{3} = 1.33 \, \text{cm}
For the 6 cm funnel: Diameter of Narrow Tube=2×12=1 cm \text{Diameter of Narrow Tube} = 2 \times \frac{1}{2} = 1 \, \text{cm}
The measures of the remaining parts for each funnel are as follows:
- For the 10 cm funnel: Total height = 13.33 cm, Height of conical part = 8.33 cm, Diameter of narrow tube = 1.67 cm. - For the 8 cm funnel: Total height = 10.67 cm, Height of conical part = 6.67 cm, Diameter of narrow tube = 1.33 cm. - For the 6 cm funnel: Total height = 8 cm, Height of conical part = 5 cm, Diameter of narrow tube = 1 cm.

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