Two pillars of equal height stand on either side of a roadway which is $150 ext{ m}$ wide. At a point in the road between the pillars, the angles of elevation of the tops of the pillars are $x^\circ$ and $y^\circ$ such that:

$$ an x^\circ = \frac{2}{5}, \quad an y^\circ = \frac{3}{5}$$

Find the height of each pillar.

Question

Two pillars of equal height stand on either side of a roadway which is $150 	ext{ m}$ wide. At a point in the road between the pillars, the angles of elevation of the tops of the pillars are $x^\circ$ and $y^\circ$ such that:

$$	an x^\circ = \frac{2}{5}, \quad 	an y^\circ = \frac{3}{5}$$

Find the height of each pillar.

Accepted Answer

**Solution:**

1. Let the height of each pillar be $h$ meters.
2. Let the point on the road between the pillars be at a distance $d$ meters from the first pillar.
3. The width of the road is $150 	ext{ m}$, so the distance from the second pillar to the point is $(150 - d)$ meters.

4. Using the tangent of the angles of elevation:
   - From the first pillar:
     $$	an x = \frac{h}{d} = \frac{2}{5} \implies h = \frac{2}{5} d$$

- From the second pillar:
     $$	an y = \frac{h}{150 - d} = \frac{3}{5} \implies h = \frac{3}{5} (150 - d)$$

5. Equate the two expressions for $h$:
   $$\frac{2}{5} d = \frac{3}{5} (150 - d)$$

6. Multiply both sides by 5 to clear denominators:
   $$2d = 3(150 - d)$$

7. Expand and simplify:
   $$2d = 450 - 3d$$

8. Bring terms involving $d$ to one side:
   $$2d + 3d = 450 \implies 5d = 450$$

9. Solve for $d$:
   $$d = \frac{450}{5} = 90 	ext{ m}$$

10. Substitute $d = 90$ into $h = \frac{2}{5} d$:
    $$h = \frac{2}{5} 	imes 90 = 36 	ext{ m}$$

**Answer:** $36 	ext{ m}$

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