Two vertical poles A and B, each of height h, are standing on opposite sides of a level road. They are 24 m apart. The point P on the road directly between the two poles is a distance x from pole A. The angle of elevation from P to the top of pole A is 60°. (a) Write h in terms of x. (b) From P the angle of elevation to the top of pole B is 30°. Find h, the height of the two poles.

Junior Cycle Mathematics Trigonometry: Two vertical poles A and B, each

Two vertical poles A and B, each of height h, are standing on opposite sides of a level road - Junior Cycle Mathematics - Question Question 1 - 2012

Question Question 1

Two vertical poles A and B, each of height h, are standing on opposite sides of a level road. They are 24 m apart. The point P on the road directly between the two p... show full transcript

Worked Solution & Example Answer:Two vertical poles A and B, each of height h, are standing on opposite sides of a level road - Junior Cycle Mathematics - Question Question 1 - 2012

Step 1

Write h in terms of x.

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Answer

To find the height h in terms of x, we will use the tangent of the angles of elevation from point P to the tops of the poles A and B.

For pole A: $\tan(60^{\circ}) = \frac{h}{x}$ Thus, $h = x \tan(60^{\circ}) = x \sqrt{3}$

For pole B: $\tan(30^{\circ}) = \frac{h}{24 - x}$ Thus, $h = (24 - x) \tan(30^{\circ}) = (24 - x) \frac{1}{\sqrt{3}}$

Equating both expressions for h: $x \sqrt{3} = (24 - x) \frac{1}{\sqrt{3}}$

So, we have:
$h = x \sqrt{3}$

Step 2

From P the angle of elevation to the top of pole B is 30°. Find h, the height of the two poles.

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Answer

Using the relationship derived previously, we'll plug in values to calculate h.

Setting up the equations from part (a): $\tan(30^{\circ}) = \frac{h}{24 - x}$ Thus, $h = (24 - x) \frac{1}{\sqrt{3}}$

Since we also found that: $h = x \sqrt{3}$

Now, substituting: $\frac{1}{\sqrt{3}} (24 - x) = x \sqrt{3}$ Multiplying through by \sqrt{3} to eliminate the fraction, we get: $24 - x = 3x$ Thus,

ightarrow x = 6 \text{ m} $$ Now we can find the height h: $$ h = 6 \sqrt{3} \text{ m} \approx 10.39 \text{ m} $$ So, the height of the poles is approximately 10.39 m.

Two vertical poles A and B, each of height h, are standing on opposite sides of a level road - Junior Cycle Mathematics - Question Question 1 - 2012

Worked Solution & Example Answer:Two vertical poles A and B, each of height h, are standing on opposite sides of a level road - Junior Cycle Mathematics - Question Question 1 - 2012

Write h in terms of x.

From P the angle of elevation to the top of pole B is 30°. Find h, the height of the two poles.

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