Multiple choice

An aeroplane is observed at the same time by two anti-aircraft batteries distant $6000m$ apart to be at elevations of ${30}^{o}$ and ${45}^{o}$ respectively. Assuming that the aeroplane is travelling directly towards the two batteries, find its height and its horizontal distance from the nearer battery.

  1. $3000(\sqrt 3+1)m$; $3000(\sqrt 3+1)m$
  2. $4000(\sqrt 2+1)m$; $3000(\sqrt 3+1)m$
  3. $3000(\sqrt 3+1)m$; $4000(\sqrt 2+1)m$
  4. $4000(\sqrt 2+1)m$; $4000(\sqrt 2+1)m$
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AI explanation

Let the height of the aeroplane be h and its horizontal distance from the nearer battery be x. Since the elevation from the nearer battery is 45 degrees, tan 45 degrees equals h divided by x, so h equals x. The batteries are 6000 m apart, so the distance from the farther battery is x plus 6000. The elevation from the farther battery is 30 degrees, giving tan 30 degrees equals h divided by (x plus 6000). Substituting h for x yields 1 divided by root 3 equals x divided by (x plus 6000), which simplifies to x root 3 equals x plus 6000. Solving for x gives x equal to 6000 divided by (root 3 minus 1), which rationalizes to 3000 times (root 3 plus 1). Therefore, both the height and the horizontal distance from the nearer battery are 3000(root 3 + 1) m.

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