Wikipedia:Reference desk/Archives/Mathematics/2022 November 5

= November 5 =

inflation and interest
If due to inflation, 1000 will be 500 after 10 years, then what should be the investment interest rate so that 1000 will become 2000 in next 10 years? 2402:3A80:1A48:7055:99CB:D871:AA1A:B7C8 (talk) 09:32, 5 November 2022 (UTC)
 * 1000 is a number and as such not subject to inflation. Perhaps you mean the price of some well-defined good, such as used to define the Big Mac index. If the annual percentage rate with which something grows is $$a\%$$, the quantity will grow in a ratio of $$100:100+a,$$ or, equivalently, by a factor of $$1+\frac{a}{100}$$ each year, and therefore by a factor of $$\left(1+\frac{a}{100}\right)^n$$ in $$n$$ years. This holds equally for investment interest rates, mortgage loan interest rates, and any other types of interest rate. So the problem is to solve the equation $$\left(1+\frac{a}{100}\right)^{10}=2$$ for $$a.$$ Taking 10th roots of both sides gives us $$1+\frac{a}{100}=\sqrt[10]{2}\approx 1.072,$$ so $$a\%\approx (1.072-1)\times 100\%=7.2\%.$$. The annual percentage rate needed to double an amount in the next 10 years is independent of other things, so the information in the first sentence seems irrelevant, but perhaps you want to know how much the growth rate should be to double the actual value while counteracting the inflation effect. If 1000&thinsp;paise buys you a Big Mac in 2022, but you will need 2000&thinsp;paise for the same thing in 2032, just doubling the nominal monetary amount means you only marked time. In 2032 you'll need 4000&thinsp;paise to buy two big Big Macs, so if you seek to quadruple the nominal amount, you need the solution to the equation with $$4$$ in the right-hand side. This comes out at $$14.9\%.$$ If you find a fund with that guaranteed rate, give us a tip. --Lambiam 13:32, 5 November 2022 (UTC)

Another sphere question
Where is it hardest to fit a sphere to the WGS84 ellipsoid when trying to maximize the farness of the nearest place with a ≥6 inch disagreement? Since ellipsoids are symmetrical this must not be a single place but everywhere in the world with the same latitude absolute value tied for worst. I suspect the easiest place is a pole with the sphere 5.999999999.. inches above the ellipsoid at the pole and sphere z-coordinate and radius picked to delay 6 inches error as long as possible. Sagittarian Milky Way (talk) 21:33, 5 November 2022 (UTC)
 * The fittability will vary between two extremes. One would expect these extremes to be attained at the poles and at the equator. At the poles, the principal curvatures coincide, which suggests an easy snug local fit with a spherical cap. At the equator, they differ maximally, which means the shape is locally hard to approximate with a spherical cap. So it seems somewhat obvious that the answer is: the equator. It may not be easy to give a watertight proof. --Lambiam 01:11, 6 November 2022 (UTC)
 * Estimating with the awesome versine function I get 0.00004999958333473664 radians of drop for a 0.01 radian arc. If this is 0.01 equatorial radians=63781.37 meters then 63781.37 meters north of the equator should be 63781.37/6335439 minimum radius of curvature radians which has a drop of 0.00005067579793915922 short radians or 0.00005033655229103247 long radians. 0.00005033655229103247-0.00004999958333473664*6378137 meters=2.149 meters, meter-level error for any compromise so lets try 0.005 radians. 79.727 meters of drop east of Null Island, 80.264 meters of drop north of Null Island. Getting close. 0.004 radians: 51.025 meters east, 51.369 meters north. 0.00375 radians!: 44.84622, 45.14847, 302.2 millimeters difference, compromise possible. Since 0.0038 radians is so small probably no complicated thing happens from radii of curvatures varying throughout the circle and it's the equator. Sagittarian Milky Way (talk) 02:31, 7 November 2022 (UTC)