This blog entry will deal with finding two formulas (approximate?) for finding the volume of a bottle. This includes plastic water bottles, beer bottles, and carry bottles.

In order to know how to find the volume, let's loom at the basic parts. This was accomplished by a basic search on Google:

Assume that the neck and body are cylinders. The shape of the shoulder is going to vary. First, let's work with a bottle with a linear shoulder, and one with a parabolic shoulder. The volume of the bottle will be measured in three parts.

V = VN + VS + VB

Where:

VN = volume of the neck

VS = volume of the shoulder

VB = volume of the body

**Bottle - Linear Shoulder: **

Neck: VN = π * r^2 * a

Body: VB = π * q^2 * c

Shoulder:

Let's use the technique of the Method of Discs.

Top Boundary: y = b

Bottom Boundary: y = 0

Left Boundary: x = 0

Right Boundary: x = (r - q)/b * y + q

The line between the points (r, b) and (q, 0).

Slope:

Δy/Δx = (b - 0)/(r - q) = b/(r - q)

Y Intercept

y = b/(r - q) * x + β

Use point (q, 0) (x = q and y = 0)

0 = b/(r - q) * q + β

β = -b/(r - q) * q

Solving for x:

y = b/(r - q) * x - b/(r - q) * q

(r - q)/b * y = b * x - b * q

x = (r - q)/b * y + q

Volume of the Shoulder:

VS =

b

∫ ((r - q)/b * y + q)^2 dy * π =

0

b

∫ (r - q)/b * ((r - q)/b * y + q)^2 dy * (π * b)/(r - q) =

0

b

[ 1/3 * ((r - q)/b * y + q)^3 ] * (π * b)/(r - q) =

0

(π * b)/(3 * (r - q)) * (r^3 - q^3) =

(π * b)/(3 * (r - q)) * (r - q) * (r^2 + r * q + q^2) =

(π * b)/3 * (r^2 + r * q + q^2)

Total Volume - Bottle: Linear Shoulder:

V = VN + VS + VB =

π * r^2 * a + (π * b)/3 * (r^2 + r * q + q^2) + π * q^2 * c

** Bottle: Parabolic Shoulder **

Neck: VN = π * r^2 * a

Body: VB = π * q^2 * c

Shoulder:

Let's use the technique of the Method of Discs.

Top Boundary: y = b

Bottom Boundary: y = 0

Left Boundary: x = 0

Right Boundary:

Parabolic Equation with roots x = -q and x = q and the curve concave downward, an equation to describe this curve can be:

y = -x^2 + q^2

y + x^2 = q^2

x^2 = q^2 - y (note we have x^2)

Volume of the Shoulder:

VS =

b

∫ q^2 - y dy * π =

0

b

[ q^2 * y - y^3/3 ] * π =

0

π * (b * q^2 - b^3/3)

Total Volume - Parabolic Shoulder:

V = VB + VS + VN =

π * q^2 * c + π * (b * q^2 - b^3/3) + π * r^2 * a

Example:

a = 1 in, b = 1 in, c = 4 in, r = 0.9 in, q = 1.5 in

V = π * 1.5^2 * 4 + π * (1 * 1.5^2 - 1^3/3) + π * 0.9^2 * 1 ≈ 36.84041 in^3

These are two ways to approximate the volume of the bottles.

Eddie

This blog is property of Edward Shore. 2014

Very interesting blog post, good job and thanks for sharing such a good blog.read more @ ball volume calculator

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