Question 25
The map shows a land border from X to Y.
State the latitude and longitude of X to the nearest degree.
[1 mark]State the latitude and longitude of Y to the nearest degree.
[1 mark]Calculate the distance between X and Y in kilometres.
[3 marks]The border runs due south, so this is a meridian distance
Watch X and Y read off a grid whose lines are 2° and 4° apart. Then see why the same longitude means no cosine: every degree along a meridian is 111.2 km.
Meridians all run pole to pole on full-size circles, so a degree along any of them is the same 111.2 km. That is why the same-meridian rule has no cosine.
Land the crosshair on each point, then pick the calculation
On the left, move the crosshair a whole degree at a time until it sits on X, then on Y. The dashed lines mark every degree the map leaves unlabelled. On the right, choose what you would type for part c). Each wrong entry is an easy slip to make.
Pick an entry. Only one of them gives the distance from X to Y.
Two readings off the map, then one rule with no cosine
Try each step yourself before you reveal it. Parts a) and b) are one mark each for a pair of coordinates. Part c) has a mark for the angle, one for the substitution and one for the distance.
Latitude first. X is just below the 2° S line. The parallels are 2° apart, so 3° S is halfway to 4° S, and X is nearer 3° S than 2° S. Then longitude. X is about three-quarters of the way from 138° E to 142° E. The meridians are 4° apart, so three-quarters of a gap is 3°.
$$138^\circ+\tfrac34\times4^\circ=141^\circ$$
The guide accepts alternative formats, such as (3° S, 141° E). Keep the S and the E, so the marker knows which hemispheres you mean.
Marker: correctly determines the latitude and longitude of X to the nearest degree
Y is about halfway between the 8° S and 10° S lines, so 9° S. The border runs straight down from X, so Y is on the same meridian, 141° E.
Marker: correctly determines the latitude and longitude of Y to the nearest degree
X and Y share a longitude, so they lie on the same meridian. The angle between them is the gap in latitude. Both are south, so subtract.
$$\begin{aligned}\text{Angular distance}&=9^\circ-3^\circ\\&=6^\circ\end{aligned}$$
Follow-through applies. If your latitudes in a) or b) were a degree out, you can still earn this mark by using your own values correctly.
Marker: determines angular distance
Write the same-meridian rule from the formula book, then substitute. There is no cosine in this one.
Marker: substitutes into appropriate rule
Evaluate. Nothing needs rounding, because 111.2 times a whole number of degrees comes out exact.
$$111.2\times6=667.2$$
The guide accepts equivalent decimal values. Give the unit, kilometres, as the question asks.
Marker: determines distance
Each phrase of the question gave you something
Two points read off a labelled grid, then one distance with a rule straight from the formula book. The 2023 subject report listed latitude and longitude, angular distance and distance between two places on Earth among the things students did well.
If I had to guess where you might lose marks, it would be choosing the rule. The same report asked teachers to train students to spot the cues that pick the right formula-book rule, and named distance between two places on Earth. The cue here is the matching longitude: same meridian, so no cosine. Putting in $\cos 3^\circ$ gives 666.29 km, close enough to look right and still the wrong rule. The other slip is the map itself. Its lines are 2° and 4° apart, so counting one gap as one degree puts X at 139° E.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2023 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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