DenominatorA — Page 313, Lesson 282

I want to walk you through a key formula for working with scale on Mercator charts. This is one of the most practical equations you'll use in navigation, so let's make sure you understand every part of it. The equation to remember is this: Denominator A over Denominator B equals cosine A over cosine B. Let me break that down. Denominator A and Denominator B are the scale denominators at two different latitudes. So if you know the scale at one latitude, you can find the scale at another. Cosine A and cosine B are simply the cosines of those two latitudes. Now, you may recall that this is also the formula used to compare the departure between two longitudes at different latitudes. Departure is the east-west distance between meridians, measured in nautical miles, and it changes with latitude because the meridians converge. The only difference now is that D, which used to represent departure in that earlier formula, now represents the denominator of scale. So the mathematical relationship is the same, but we're applying it to scale instead of distance. Let's work through an example to see how this works in practice. Example 3: On a Mercator chart, the scale at 54° South is 1:2,000,000. What is the scale at 25° North? The options are: a. 1:2,000,000 b. 1:3,084,000 c. 1:1,121,000 d. 1:3,825,000 First, use your knowledge to discount answers a and c. Here's the reasoning: at 25° North, you are closer to the Equator than at 54° South. On a Mercator chart, scale expands as you move away from the Equator and contracts as you move toward it. So as we go from 54° South to 25° North, we are moving toward the Equator, meaning the scale must contract. When scale contracts, the denominator gets larger. So the denominator at 25° North must be larger than 2,000,000. That immediately rules out a, which is the same, and c, which is smaller. In some cases, there may only be one appropriate answer given and you can spot it without calculation. Here, answers b and d could both be correct, so we need to use the formula. Let's set it up. Let D subscript A be the scale denominator at 25° North, and D subscript B be the scale denominator at 54° South, which is 2,000,000. The formula is: D subscript A over 2,000,000 equals cosine of 25° over cosine of 54°. To solve for D subscript A, we multiply both sides by 2,000,000: D subscript A equals cosine 25° times 2,000,000, all divided by cosine 54°. Now, cosine of 25° is approximately 0.9063, and cosine of 54° is approximately 0.5878. So: D subscript A equals 0.9063 times 2,000,000 divided by 0.5878, which gives us approximately 3,083,806. Rounding that, the scale at 25° North is 1:3,084,000, which is answer b. Now let's look at a different type of problem: given a fixed chart distance between meridians, find the Mercator scale at a specific latitude. Example 4: On a Mercator chart, the chart length between two meridians, 160° East and 160° West, is 30 centimetres at 30° South. What is the scale of the chart at 30° South? There are two ways of doing this, and there is very little to choose between them. The first method is to treat it as a departure problem. Remember, scale equals chart length divided by earth distance. The chart length is given as 30 centimetres. The earth distance is the departure, which is the actual distance on the Earth's surface between those two meridians at that latitude. Departure equals change in longitude in minutes times cosine of the latitude. The change in longitude from 160° East to 160° West is 40 degrees. Why 40? Because going from 160° East to 180° is 20 degrees, and from 180° to 160° West is another 20 degrees, giving a total of 40 degrees. We convert that to minutes by multiplying by 60, so 40 degrees times 60 equals 2,400 minutes. Then we multiply by the cosine of the latitude. At 30° South, cosine of 30° is 0.866. So departure equals 2,400 minutes times 0.866, which gives us approximately 2,078 nautical miles. So the earth distance is 2,078 nautical miles. The chart length is 30 centimetres. To find the scale, we need both measurements in the same units, but for now, you can see the principle: scale at that latitude is chart length divided by earth distance. We'll complete that calculation in a moment, but first, make sure you understand that the departure formula gives us the actual distance on the Earth's surface between those two meridians at that specific latitude.