BBC Bitesize - GCSE Maths - Units of measure - Edexcel - Revision 7
Visual comparison of a kilometre, statute mile, and nautical mile. This article is about distance measurement at sea. For distance measurement on land, see mile. For the US unit based on the equator, see geographical mile. A nautical mile is a unit of measurement used in both air and marine navigation, and for the . Help · About Wikipedia · Community portal · Recent changes ·. Convert between miles and kilometres teaching resources for National Curriculum Resources. Created for teachers, by teachers! Professional. If you need to use an imperial to metric conversion in an exam, the values will be given to you. Two common conversions are: 5 miles ≈ 8 kilometres; 1 inch.
The standardised Austrian mile used in southern Germany and the Austrian Empire was 7. Earlier values had varied: The Germans also used a longer version of the geographical mile. Afterthe Ottoman mile was replaced with the modern Turkish mile 1, The old Imperial value of the yard was used in converting measurements to metric values in India in a Act of the Indian Parliament.
When the international mile was introduced in English-speaking countries, the basic geodetic datum in America was the North American Datum of NAD In the United States, statute mile normally refers to the survey mile,  about 3.
While most countries replaced the mile with the kilometre when switching to the International System of Unitsthe international mile continues to be used in some countries, such as LiberiaMyanmar the United Kingdom  and the United States. Mariana Islands,  Samoa,  St. But in some cases, such as in the U. State Plane Coordinate Systems SPCSswhich can stretch over hundreds of miles,  the accumulated difference can be significant, so it is important to note that the reference is to the U.
Mile - Wikipedia
The United States redefined its yard inbut this resulted in U. State Plane Coordinate Systems were then updated, but the National Geodetic Survey left individual states to decide which if any definition of the foot they would use. Survey feet and one in international feet. Twenty-four states have legislated that surveying measures be based on the U. Each circle shown is a great circle —the analog of a line in spherical trigonometry—and hence the shortest path connecting two points on the globular surface.
Meridians are great circles that pass through the poles. Nautical mile The nautical mile was originally defined as one minute of arc along a meridian of the Earth. Using the WGS84 ellipsoidthe commonly accepted Earth model for many purposes today, one minute of latitude at the WGS84 equator is 6, feet and at the poles is 6, The average is about 6, feet about 1, metres or 1. In the United States, the nautical mile was defined in the 19th century as 6, Other nations had different definitions of the nautical mile, but it is now internationally defined to be exactly 1, metres 6, Nautical miles and knots are almost universally used for aeronautical and maritime navigation, because of their relationship with degrees and minutes of latitude and the convenience of using the latitude scale on a map for distance measuring.
The data mile is used in radar -related subjects and is equal to 6, feet 1. Thus, the radar statute mile is Geographical mile The geographical mile is based upon the length of a meridian of latitude.
Typically the largest streets are about a mile apart, with others at smaller intervals. In the Manhattan borough of New York City "streets" are close to 20 per mile, while the major numbered "avenues" are about six per mile.
Related rates: Approaching cars
So we know that x is equal to 0. What is the rate at which x is changing with respect to time?
Well, we know it's 30 miles per hour is how fast we're approaching the intersection, but x is decreasing by 30 miles every hour. So we should say it's negative 30 miles per hour. So we know what y is. We know what x is.
We know how fast y is changing, how fast x is changing with respect to time. So what we could try to do here is come up with a relationship between x, y, and s. And then differentiate that relationship with respect to time. And it seems like we have pretty much everything we need to solve for this. So what's a relationship between x, y, and s? Well we know that this is a right triangle. The streets are perpendicular to each other. So we can use the Pythagorean theorem. We know that x squared plus y squared is going to be equal to s squared.
And then we can take the derivative of both sides of this with respect to time to get a relationship between all the things that we care about. So what's the derivative of x squared with respect to time?
Well, so you're going to need the derivative of x squared with respect to x, which is just 2x, times the derivative of x with respect to time, times dx dt. Once again, just the chain rule. Derivative of something squared with respect to the something, times the derivative of the something with respect to time. And we use similar logic right over here when we want to take the derivative of y squared with respect to time.
Derivative of y squared with respect to y, times the derivative of y with respect to time. Now on the right-hand side of this equation, we once again take the derivative with respect to time. So it's the derivative of s squared with respect to s, which is just 2s. Times the derivative of s with respect to time. Once again, this is all just an application of the chain rule. So now it looks like we know what x is, we know what dx dt is, we know what y is, we know what dy dt is.
All we need to figure out is what s and then what ds dt is, the rate at which this distance is changing with respect to time. Well what's s right now? Well we can actually use the Pythagorean theorem at this exact moment.
We know that x squared-- so x is 0. Well this is 0. This is 1, it is equal to s squared. And we only care about positive distances, so we have s is equal to 1 right now. So we also know what s is. So let's substitute all of these numbers in and then try to solve for what we came here to do.
Solve for ds dt.