9/21/2023 0 Comments Target measuring tape![]() ![]() ![]() Estimates are commonly used on farms for management purposes, as well as for determining the dosage of medications, including antibiotics and anthelmintics. ![]() The accurate weighing of pre‐weaned calves is important for measuring the growth rate of calves. 2 Studies have also highlighted there are additional health and production benefits associated with growing at the target DLWG of 0.8 kg/day in the pre‐weaning period, for example, the average DLWG in the first 2 months of life has been linked to survival to the end of the second lactation and increased milk production. Factors affecting the DLWG of heifers and the use of different management decisions to maintain target DLWG while reducing the cost per kg of gain have been the subject of a growing amount of research, especially the pre‐weaning period where the cost is on average the highest at £3.14 per day on UK farms. 1 Feed costs represent a significant financial input, which may be affected by the desired target daily liveweight gain (DLWG) for the heifer. I inadvertently ended up writing three blog posts in a row related to measuring tapes.Heifer rearing represents a significant financial investment to dairy farmers, costing on average £1,391 on farms in the United Kingdom. But if we could pull tight enough to limit the sag to 1 inch, the measurement error would be more like 1/10 of an inch. If instead we measure from (0, 4) to (120, 0) with a tape that sags like a parabola touching (60, 0), then the error will be closer to an inch and a half. If we measure with a straight line from (0, 0) to (120, 4) instead, there will be an error of about 1/15 of an inch. Going back to our example of measuring a 10 ft (120 inch) room, we want to measure from (0, 0) to (120, 0). If y is small relative to x, the error is still small, but about 20x larger than before. So the error is on the order of (32/3) y²/ x, where as in the earlier post the error was on the order of (1/2) y²/ x. The equation of the parabola passing through our three specified points, as a function of t, isĮven with our simplifying assumption that the tape bends like a parabola, the arclength calculation is still a little tedious, but if I’ve done things correctly the result turns out to be (More on how well a parabola approximates a catenary here.) Since we are assuming the amount of sag y is small relative to the horizontal distance x, a parabola will do just as well as a catenary. The calculations depend on solving for a scaling factor, and that calculation cannot be done in closed form. But the resulting calculations are too complicated. My first thought was to assume our tape measure takes the shape of a catenary, because that’s the shape of a handing cable. About how large will our error be as a function of y? As before, we’ll assume y is small relative to x. We end exactly at our target, but the tape bends. In this post we will consider measuring from (0, y) to ( x, y), but with the tape measure sagging to ( x/2, 0) in the middle. We assumed the tape measure was straight, but didn’t aim straight at the target. You’d like to measure from (0, 0) to ( x, 0), but something is in the way, and so you measure from (0, 0) to ( x, y), where y is small relative to x. Yesterday I wrote a blog post about not measuring straight toward your target. There are a couple ways in which a measurement might not be straight. ![]()
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