To calculate the volume breakdown of the four layers within the AIDA funnel modeled as an equilateral triangle, divided equally, let’s approach this systematically:
Step 1: Geometry of the Funnel
An equilateral triangle has three equal sides, each 1 meter (100 cm, or 1000 mm). The funnel can be represented as a triangular pyramid, known as a tetrahedron, with the triangular base perpendicular to the triangular sides. Dividing this funnel into four layers essentially divides its height proportionately.
Step 2: Dimensions and Calculations
- Base Area: The area of an equilateral triangle with sides a=1a = 1 meter is:
Area=34⋅a2=34⋅(1)2=34≈0.433 m2.\text{Area} = \frac{\sqrt{3}}{4} \cdot a^2 = \frac{\sqrt{3}}{4} \cdot (1)^2 = \frac{\sqrt{3}}{4} \approx 0.433 \text{ m}^2.
- Height: The height of an equilateral triangle (or the tetrahedron itself) can be calculated using:
h=32⋅a=32⋅1=32≈0.866 m.h = \frac{\sqrt{3}}{2} \cdot a = \frac{\sqrt{3}}{2} \cdot 1 = \frac{\sqrt{3}}{2} \approx 0.866 \text{ m}.
Step 3: Total Volume of the Funnel
The volume of a tetrahedron is:
Volume=13⋅Base Area⋅Height=13⋅0.433⋅0.866≈0.125 m3.\text{Volume} = \frac{1}{3} \cdot \text{Base Area} \cdot \text{Height} = \frac{1}{3} \cdot 0.433 \cdot 0.866 \approx 0.125 \text{ m}^3.
Step 4: Dividing into Four Layers
Since the funnel is divided equally into four layers (in height), the volumes of the layers will not be equal due to the pyramidal shape. The formula for the volume of a pyramid section is proportional to the cube of its height. If the total height is divided into four equal segments, the cumulative volumes are calculated as follows:
- Layer 1 (topmost): Volume ratio = 1343=164\frac{1^3}{4^3} = \frac{1}{64}
- Layer 2: Volume ratio = 23−1364=764\frac{2^3 – 1^3}{64} = \frac{7}{64}
- Layer 3: Volume ratio = 33−2364=1964\frac{3^3 – 2^3}{64} = \frac{19}{64}
- Layer 4 (bottommost): Volume ratio = 43−3364=3764\frac{4^3 – 3^3}{64} = \frac{37}{64}.
Step 5: Layer Volumes
Multiply each ratio by the total volume 0.125 m30.125 \, \text{m}^3:
- Layer 1: 0.125⋅164/0.00195 M3
- Layer 2: 0.125⋅764/0.01367 M3
- Layer 3: 0.125⋅1964/0.03711 M3
- Layer 4: 0.125⋅37640.07227 M3
Final Volume Breakdown
| Layer | Volume (approx) |
|---|---|
| Layer 1 | 0.00195 m3 – 1/64 |
| Layer 2 | 0.01367 m3 – 7/64 |
| Layer 3 | 0.03711 m3 – 19/64 |
| Layer 4 | 0.07227 m3 – 37/64 |
Volume Breakdown in Proportions
If the total height is evenly split into four, the volumes of the layers increase proportionally based on the cube of their relative heights:
- Layer 1 (Bottom): Smallest slice, approximately 1.6% of the total volume.
- Layer 2: Approximately 10.9% of the total volume.
- Layer 3: Approximately 29.7% of the total volume.
- Layer 4 (Top): Largest slice, approximately 57.8% of the total volume.
