How change in length affects area and volume

By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

14 July 2024

In Kenyan parlance, a parcel of land measuring 50ft by 100ft is called an eighth of an acre. I have explained in a past article that this is incorrect: the correct measurement should be about 52ft by 104ft, or, better still, 54ft-6in by 100ft. In addition, most of the plots that are sold as “50 x 100” are actually surveyed as 15m by 30m. This translates to 49ft by 98ft. So, in short, it is neither an eighth of an acre nor is it a 50ft by 100ft!

Now, if you get a plot that is 98ft long instead of 100ft, it means that you have been short-changed by 2 per cent on the length. The width of 49ft instead of 50ft is also 2 per cent shorter that what is advertised. The question then is: how much area are you losing?

It is easy to calculate: 50 x 100 = 5,000 square feet; on the other hand, 49 x 98 = 4,802sq.ft. The difference between the two areas is 198sq.ft. This is a loss of 3.96 per cent – almost 4per cent. In other words, the percentage loss of area is almost double that of the length.

This should not be surprising. If you divide each side of a square by half, you end up with four quarters – the reduction in area (a factor of four) is double the reduction in the lengths (halves). If you change all the lengths of the sides by a certain factor, the enclosed area changes by the square of that factor.

In the case of the space enclosed by a three-dimensional solid, the change in the volume is equal to the cube of the change in the lengths. For example; the volume of a cube measuring 10cm x 10cm x 10cm is 1,000cc (one litre). If you divide each side by half, the new volume will be 125cc – an eight of the original! The lengths of the sides have gone down by a factor of two, so, the volume goes down by a factor of 8 (; 2 x 2 x 2 = 8) from 1000cc to 125cc.

I recently spent some time explaining this idea to a medical radiologist. He had taken a scan on a patient and found a growth measuring 6.1cm x 5.4cm x 3.2cm. After six months of medication, the patient returned and the new measurements were 6.0cm x 5.2cm x 2.9cm. At first glance, the reduction appears insignificant.

Even though the lump is not a cube, we can get information about change in volume by multiplying the dimensions. Thus; it started at 105.4 and, after treatment, it is now 90.5. This is a 14 per cent change in the volume. I am not a medical doctor but that does not seem like an insignificant reduction.