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Posted on December 15, 2008 @ 09:11:00 AM by Paul Meagher
This blog is part of a blog series:
This week I am going to discuss some recreational math I've been doing on the topic of "Calculating Loads".
It all started about 3 weeks ago when we got a heavy snowfall. I started shovelling the snow and decided to take some measurements of my driveway dimensions to see if I could figure out the total weight of the snow I was displacing from the paved driveway to the lawn area adjacent the driveway.
I measured the snow to be about 10 inches high and of fairly uniform thickness throughout the driveway. I estimated the unit weight of the snow to be about 20 lbs per cubic foot (the snow was damp and compacted). Later I learned that one method of measuring the weight would have been to outline the 2D area of a cubic foot in the snow, record the height of the snow, shovel the snow into a garbage bag, and measure the weight of the garbage bag. Perhaps I'll try this method on another snow removal day. It will give me something to think about whilst I am shovelling.
My driveway has an irregular shape and to simplify the calculations I used a trapezoidal shape to approximate its dimensions: 19 feet wide at the driveway entrance and 33 feet wide at the end which is marked by our garage entrance. I didn't measure the length of the driveway but estimated it to be about 100 feet long. With all these measurements and estimates in place, I was ready to develop a php-based script called snow_load.php to calculate a total volume and weight of snow on my driveway, aka, the snow load. The snow load calculation would give me a sense of how much of a workout it was to displace the snow off the driveway.
<?php
/**
* @script snow_load.php
* @author Paul Meagher
*
* @see http://argyll.epsb.ca/jreed/math9
* /strand3/trapezoid_area_per.htm
*/
// Using foot units
$width_1 = 33;
$width_2 = 19;
$length = 100;
// Ten inches of snow can be expressed in foot
// units this way
$height = 10/12;
// Weight of snow per cubic foot volume
$lbs_per_cubic_foot = 20;
// First compute area using trapezoid area
// formula
$area = 0.5 * $length * ($width_1 + $width_2);
// Next multiply area by height to get cubic foot
// volume
$volume = $area * $height;
// Total weight of snow removed
$weight = $volume * $lbs_per_cubic_foot;
// Number of decimal places for output
$decimals = 0;
?>
<p>
Volume of snow is
<?php echo number_format($volume, $decimals) ?>
ft<sup>3</sup>.
</p>
<p>
Total weight of snow removed was
<?php echo number_format($weight, $decimals) ?>
lbs.
</p>
When I pointed my browser at the webserver where the script resided, it generated the following output:
Volume of snow is 2,167 ft3.
Total weight of snow removed was 43,333 lbs.
It was at this point that I should have left good-enough alone and accepted this result, instead I recreated on the idea of attaining more accuracy in my calculation by approximating the shape of the driveway with a polygon. Tomorrow I will talk about that calculation when I discuss a generalized SurfaceLoad.php class. After that I will discuss some visualization work I did which I packaged into a DrawPolygon.php class.
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