Explanation:

Let's break down the problem:

- Let the height of the tree be *h* meters.

- Let the distances from the foot of the tree to posts A and B be *x* and *x + 4* meters.

- From the top of the tree, angles of depression are 45° (to A) and 60° (to B).

Using tan(?) = opposite/adjacent:

- For post A: tan 45° = h / x ? h = x

- For post B: tan 60° = h / (x + 4) ? v3 = h / (x + 4) ? h = v3(x + 4)

Set equal since h = x:

x = v3(x + 4)

x = v3x + 4v3

x - v3x = 4v3

x(1 - v3) = 4v3

x = 4v3 / (1 - v3)

Rationalize denominator:

x = 4v3(1 + v3) / [(1 - v3)(1 + v3)]

= 4v3(1 + v3) / (1 - 3)

= 4v3(1 + v3) / (-2)

= -2v3(1 + v3)

But *x* must be positive; take modulus:

x = 2v3(v3 + 1)

So, h = x = 2(v3 + 1) meters

Summary of options:

- Option 1: v3 + 1 (too small, not correct)

- Option 2: v3(v3 + 1) (missing factor of 2, not correct)

- Option 3: 2(v3 + 1) correct value

- Option 4: 4v3(v3 + 1) (too large)

Final answer:

Option 3: 2(v3 + 1) — this is the correct answer!