Two spotters of tornado are on a road running east-west and are 35 miles apart. The west man spots a tornado at bearing N 43° E and the east man spots the same tornado on a bearing of N 59° W. How far is the tornado from each man and how far is the tornado from the road?

The distance between the tornado and the spotter at the west is 30.67 miles
The distance between the tornado and the spotter at the east is 24.40 miles
The distance between the tornado and the road is 20.91 miles

See attachment for the positioning of the tornado, and the two spotters.

First, we calculate the angle T.

[tex]\mathbf{\angle T + 43 + 59 = 180}[/tex] ---- Sum of angles in a triangle

[tex]\mathbf{\angle T +102 = 180}[/tex]

Subtract 102 from both sides

[tex]\mathbf{\angle T = 78}[/tex]

From the attached figure:

y represents the distance between the tornado and the spotter at the west
z represents the distance between the tornado and the spotter at the east
x represents the distance between the tornado and the road

To calculate y, we make use of the following sine formula

[tex]\mathbf{\frac{y}{sin(59)} = \frac{35}{sin(78)}}[/tex]

Make y the subject

[tex]\mathbf{y = \frac{35}{sin(78)} \times sin(59)}[/tex]

[tex]\mathbf{y = 30.67}[/tex]

To calculate z, we make use of the following sine formula

[tex]\mathbf{\frac{z}{sin(43)} = \frac{35}{sin(78)}}[/tex]

Make z the subject

[tex]\mathbf{z = \frac{35}{sin(78)} \times sin(43)}[/tex]

[tex]\mathbf{z = 24.40}[/tex]

Next, we calculate x, using the following sine trigonometry ratio

[tex]\mathbf{sin(59) = \frac{x}{z}}[/tex]

Make x the subject

[tex]\mathbf{x = z \times sin(59)}[/tex]

Substitute [tex]\mathbf{z = 24.40}[/tex]

[tex]\mathbf{x = 24.40 \times sin(59)}[/tex]

[tex]\mathbf{x = 20.91}[/tex]

Hence, the required distances are 30.67 miles, 24.40 miles and 20.91 miles.

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