Tuesday, September 18, 2012

Volume of a Parabolic Vase and Casio fx-78

Volume of a Parabolic Vase

Introduction

The following procedure is how to find the volume of a parabolic vase. This is a vase where the "sides" (for the lack of a better term) form a parabolic curve. Each side of a vase has a smooth bump, away from the center, with one peak. Imagine if that you can turn a vase on its side. The result is like the illustration above.

Note: The following procedure works with vases where the "sides" curved towards the center, rather than away.

Steps

Using calculus, we can approximate the the volume by taking the following steps:

1. Measure the length of the vase. Let x be this length.
2. Find the "center line" of the vase. From the center line, find the distance from the center line to the end of the case. Do this for both ends of the vase, and at half of the length. Get the best measurement you can. From one end to the other, these are lengths y_0, y_1, and y_2, respectively.
3. Use the data to form the parabolic curve. Find a, b, and c.


4. With a, b, and c determined, plug in a, b, c, and x and calculate the following integral.


Most graphing calculators can handle this task easily.

Note: If there is an official term for these type of vases, I would like to know. Many thanks in advance.

Example

Find the volume of a parabolic vase 4 inches long with the following: (see the illustration "Parabolic Vases with measurements").


So we have these points to form the parabolic equation with:
(0,4), (2,6), (4,4)

Then:

which leads to:

c = 4
b = 2
a = -0.5

The integral is:



Calculating the integral gives an approximate area of 134.041287 in^2.


Found at the Swap Meet: Casio fx-78

Several Sundays ago I purchased this calculator at the Azusa Swap Meet. This calculator, made in 1982, is not only basic, but quite small. It still works.

Have a great day, Eddie


This blog is property of Edward Shore. © 2012

3 comments:

  1. Hi,
    how exactly do you find a, b and c? I really have no idea how to calculate within those square parentheses.
    Thank you.

    ReplyDelete
    Replies
    1. Meggie,

      It is not parenthesis, it is matrix multiplication.

      To find c, b, and a respectively, first take the inverse of the matrix [[1, 0, 0][1, x/2, x^2/4][1, x, x^2]] then multiply the result by the matrix [[y0] [y1] [y2]]. You will get a matrix of 3 rows, 1 column. The first row corresponds to c, the second row corresponds to b, the third row corresponds to a.

      Hope that helps,

      Eddie

      Delete

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