Great Circle Distances

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Question 1 of 8

1. Question
Find the position of $Y$ $Y$ relative to $X$ $X$ given the following coordinates:

$X (25 ° N, 150 ° E)$ $X(25°N,150°E)$

$Y (15 ° S, 150 ° W)$ $Y(15°S,150°W)$
- 1. $Y$ $Y$ is $60 °$ $60°$ East and $40 °$ $40°$ South of $X$ $X$
- 2. $Y$ $Y$ is $40 °$ $40°$ West and $80 °$ $80°$ South of $X$ $X$
- 3. $Y$ $Y$ is $100 °$ $100°$ East and $60 °$ $60°$ North of $X$ $X$
- 4. $Y$ $Y$ is $60 °$ $60°$ West and $40 °$ $40°$ North of $X$ $X$
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The prime meridian is a line of longitude at $0 °$ $0°$ .

On the opposite side of the prime meridian lies the $180 °$ $180°$ meridian.

First, form a scale using the top view of the globe

Count starting from the longitude of $X$ $X$ to the longitude of $Y$ $Y$

$X (25 ° N,$ $X(25°N,$ $150 ° E$ $150°E$ $)$ $)$

$Y (15 ° S,$ $Y(15°S,$ $150 ° W$ $150°W$ $)$ $)$

From the scale above, we know that $Y$ $Y$ is $60 °$ $60°$ East of $X$ $X$

Lastly, add the latitude of $X$ $X$ and $Y$ $Y$

$X ($ $X($ $25 ° N$ $25°N$ $, 150 ° E)$ $,150°E)$

$Y ($ $Y($ $15 ° S$ $15°S$ $, 150 ° W)$ $,150°W)$

$25 ° + 15 °$ $25°+15°$ $=$ $=$ $40 °$ $40°$

Hence, $Y$ $Y$ is $40 °$ $40°$ South of $X$ $X$

$Y$ $Y$ is $60 °$ $60°$ East and $40 °$ $40°$ South of $X$ $X$

Question 2 of 8

Find the shortest distance between San Francisco and Seattle given the following coordinates:

San Fracisco

(38 ° N, 122 ° W)

$(38°N,122°W)$

Seattle

(47 ° N, 150 ° W)

$(47°N,150°W)$

Round your answer to the nearest whole number

(1005)km

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Arc Length Formula

L = \frac{θ}{360} \times 2 π r

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are in the same longitude, meaning both of them lie on the same circle.

Draw the sector that covers the distance from Seattle to San Francisco.

We also know that the radius of earth is $6400$ $6400$ km

Find $θ$ $theta$ by getting the difference between the two latitudes

San Fracisco

(38 ° N, 122 ° W)

$(38°N,122°W)$

Seattle

(47 ° N, 150 ° W)

$(47°N,150°W)$

$θ$ $theta$	$=$ $=$	$47 ° - 38 °$ $47°-38°$
$θ$ $theta$	$=$ $=$	$9 °$ $9°$

Lastly, substitute known values into the Arc Length Formula

θ = 9 °

$theta=9°$

r = 6400

$r=6400$ km

$L$ $L$	$=$ $=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$ $=$	$\frac{\color{#00880A}{9}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$ $=$	$\frac{1}{40} (40212.3)$ $1/40(40212.3)$

	$=$ $=$	$1005$ $1005$ km

Hence, the shortest distance between Seattle and San Francisco is

1005

$1005$ km

1005

$1005$ km

Question 3 of 8

Find the shortest distance between Melbourne and Cairns given the following coordinates:

Melbourne

(38 ° S, 145 ° E)

$(38°S,145°E)$

Cairns

(17 ° S, 145 ° E)

$(17°S,145°E)$

Round your answer to the nearest whole number

(2346)km

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Arc Length Formula

L = \frac{θ}{360} \times 2 π r

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are in the same longitude, meaning both of them lie on the same circle.

Draw the sector that covers the distance from Melbourne to Cairns.

We also know that the radius of earth is $6400$ $6400$ km

Find $θ$ $theta$ by getting the difference between the two latitudes

Melbourne

(38 ° S, 145 ° E)

$(38°S,145°E)$

Cairns

(17 ° S, 145 ° E)

$(17°S,145°E)$

$θ$ $theta$	$=$ $=$	$38 ° - 17 °$ $38°-17°$
$θ$ $theta$	$=$ $=$	$21 °$ $21°$

Lastly, substitute known values into the Arc Length Formula

θ = 21 °

$theta=21°$

$r=6400$ km

$L$	$=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$	$\frac{\color{#00880A}{21}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$	$2345.7$

	$=$	$2346$ km

Hence, the shortest distance between Melbourne and Cairns is

$2346$ km

Question 4 of 8

Find the shortest distance between Adelaide and Tokyo given the following coordinates:

Adelaide

$(35°S,139°E)$

Tokyo

$(35°N,139°E)$

Round your answer to the nearest whole number

(7819)km

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Arc Length Formula

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are in the same longitude, meaning both of them lie on the same circle.

Draw the sector that covers the distance from Adelaide to Tokyo.

We also know that the radius of earth is $6400$ km

Find $theta$ by adding the two angle measures

$theta$	$=$	$35°+35°$
$theta$	$=$	$70°$

Lastly, substitute known values into the Arc Length Formula

$theta=70°$

$r=6400$ km

$L$	$=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$	$\frac{\color{#00880A}{70}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$	$7819.08$

	$=$	$7819$ km

Hence, the shortest distance between Adelaide and Tokyo is

$7819$ km

Question 5 of 8

Find the shortest distance between Stockholm and Cape Town given the following coordinates:

Stockholm

$(59°N,18°E)$

Cape Town

$(34°S,18°E)$

Round your answer to the nearest whole number

(10388)km

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Arc Length Formula

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are in the same longitude, meaning both of them lie on the same circle.

Draw the sector that covers the distance from Stockholm to Cape Town.

We also know that the radius of earth is $6400$ km

Find $theta$ by adding the two angle measures

$theta$	$=$	$59°+34°$
$theta$	$=$	$93°$

Lastly, substitute known values into the Arc Length Formula

$theta=93°$

$r=6400$ km

$L$	$=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$	$\frac{\color{#00880A}{93}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$	$10388.2$

	$=$	$10388$ km

Hence, the shortest distance between Stockholm and Cape Town is

$10388$ km

Question 6 of 8

Find the shortest distance between these two points on the equator:

$A (0°,18°W)$

$B (0°,21°E)$

Round your answer to the nearest whole number

(4356)km

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Arc Length Formula

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are on the equator, meaning both of them lie on the same circle.

Draw the sector that covers the distance from

$A$ to

$B$ .

We also know that the radius of earth is $6400$ km

Find $theta$ by adding the two angle measures

$theta$	$=$	$18°+21°$
$theta$	$=$	$39°$

Lastly, substitute known values into the Arc Length Formula

$theta=39°$

$r=6400$ km

$L$	$=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$	$\frac{\color{#00880A}{39}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$	$4356.34$

	$=$	$4356$ km

Hence, the shortest distance between

$A$ and

$B$ is

$4356$ km

Question 7 of 8

Find the shortest distance between these two points on the equator:

$Q (0°,95°W)$

$P (0°,120°E)$

Round your answer to the nearest whole number

(16197)km

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Arc Length Formula

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are on the equator, meaning both of them lie on the same circle.

Draw the sector that covers the distance from

$Q$ to

$P$ .

Remember that we are looking for the shortest distance, so

$theta$ should be the angle subtended by the shorter arc

Find $theta$ by subtracting the two angle measures from

$360°$

$theta$	$=$	$360°-95°-120°$
$theta$	$=$	$145°$

Lastly, substitute known values into the Arc Length Formula

$theta=145°$

$r=6400$ km

$L$	$=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$	$\frac{\color{#00880A}{145}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$	$16196.7$

	$=$	$16197$ km

Hence, the shortest distance between

$Q$ and

$P$ is

$16197$ km

Question 8 of 8

Find the shortest distance between Nairobi and Singapore given the following coordinates:

Nairobi

$(0°,95°W)$

Singapore

$(0°,120°E)$

Round your answer to the nearest whole number

(7596)km

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Arc Length Formula

$L=\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$

Notice that the two coordinates are on the equator, meaning both of them lie on the same circle.

Draw the sector that covers the distance from Nairobi to Singapore.

Remember that we are looking for the shortest distance, so

$theta$ should be the angle between Nairobi and Singapore

Find $theta$ by getting the difference of the two angle measures

$theta$	$=$	$104°-36°$
$theta$	$=$	$68°$

Lastly, substitute known values into the Arc Length Formula

$theta=68°$

$r=6400$ km

$L$	$=$	$\frac{\color{#00880A}{\theta}}{360}\times2\pi\color{#9a00c7}{r}$	Arc Length Formula

	$=$	$\frac{\color{#00880A}{68}}{360}\times2\pi(\color{#9a00c7}{6400})$	Substitute values

	$=$	$7595.67$

	$=$	$7596$ km

Hence, the shortest distance between Nairobi and Singapore is

$7596$ km

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