A tower stands at the end of a straight road. From two points on the road 500 m apart, the angles of elevation to the top are 45° (farther point) and 60° (nearer point). Find the height of the tower (in exact form).

Introduction / Context:
Two elevation angles from collinear points separated by known distance allow solving for both the tower height and the nearer distance by simultaneous tangent equations.

Given Data / Assumptions:

• Let x be the nearer distance to the tower (angle 60°); farther distance = x + 500 with angle 45°.
• Height h unknown.

Concept / Approach:
Use tan 60° = h/x and tan 45° = h/(x + 500). Equate h from both to eliminate h and solve for x, then substitute to get h in exact surd form.

Step-by-Step Solution:

Verification / Alternative check:
Rationalize: h = 500√3(√3 + 1)/( (√3 − 1)(√3 + 1)) = 500(3 + √3)/2 ≈ 1,116.0 m; plugging back reproduces 45° and 60°.

Why Other Options Are Wrong:
Using √3 + 1 in denominator flips the comparative sizes; 5000√3 is dimensionally off; the provided exact form in option (a) is correct.

Common Pitfalls:
Mixing which point is nearer/farther; algebra sign errors when isolating x; skipping rationalization can hide mistakes.

Final Answer:
500√3 / (√3 − 1)