We are given that the angle of elevation to the top of a tree changes as we move closer. First, we measure the angle of elevation as $30^\circ$, and then after walking 20 feet closer, the angle becomes $45^\circ$. We are tasked with finding the exact height of the tree.

Step-by-Step Solution

1. Define Variables:

   • Let $h$ be the height of the tree.
   • Let $d$ be the initial distance from the tree when the angle of elevation is $30^\circ$.

2. Apply Trigonometry: From the point where the angle of elevation is $30^\circ$, we can set up a relationship using the tangent of the angle: $\tan(30^\circ) = \frac{h}{d}$ Since $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, this gives: $\frac{1}{\sqrt{3}} = \frac{h}{d}$ Solving for $h$, we get: h = \frac{d}{\sqrt{3}} \tag{1}

3. Use the New Position: After walking 20 feet closer, the distance to the tree becomes $d - 20$ feet, and the angle of elevation is $45^\circ$. Using the tangent of $45^\circ$, we get: $\tan(45^\circ) = \frac{h}{d - 20}$ Since $\tan(45^\circ) = 1$, this simplifies to: $1 = \frac{h}{d - 20}$ Solving for $h$, we have: h = d - 20 \tag{2}

4. Set Up a System of Equations: From equation (1), we know that $h = \frac{d}{\sqrt{3}}$, and from equation (2), $h = d - 20$. Equating the two expressions for $h$, we get: $\frac{d}{\sqrt{3}} = d - 20$

5. Solve for $d$: Multiply both sides of the equation by $\sqrt{3}$ to eliminate the fraction: $d = \sqrt{3}(d - 20)$ Expanding the right-hand side: $d = \sqrt{3}d - 20\sqrt{3}$ Now, bring all terms involving $d$ to one side: $d - \sqrt{3}d = -20\sqrt{3}$ Factor out $d$: $d(1 - \sqrt{3}) = -20\sqrt{3}$ Solving for $d$: $d = \frac{-20\sqrt{3}}{1 - \sqrt{3}}$

   To rationalize the denominator, multiply the numerator and denominator by $1 + \sqrt{3}$: $d = \frac{-20\sqrt{3}(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})}$ Simplifying the denominator using the difference of squares: $d = \frac{-20\sqrt{3}(1 + \sqrt{3})}{1^2 - (\sqrt{3})^2} = \frac{-20\sqrt{3}(1 + \sqrt{3})}{1 - 3} = \frac{-20\sqrt{3}(1 + \sqrt{3})}{-2}$ Simplifying the expression: $d = 10\sqrt{3}(1 + \sqrt{3}) = 10\sqrt{3} + 30$

6. Find the Height of the Tree: Using equation (2), we now find $h$. Since $h = d - 20$, substitute $d = 10\sqrt{3} + 30$: $h = (10\sqrt{3} + 30) - 20 = 10\sqrt{3} + 10$

Thus, the exact height of the tree is $\boxed{10\sqrt{3} + 10}$ feet.

Would you like to go over any particular part of the solution, or do you have further questions?

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Here are 5 related questions:

1. What is the height of the tree if the second angle of elevation were different?
2. How would the solution change if the tree were on a slope?
3. Can you solve for the distance to the tree using another method?
4. How would the problem change if both distances and angles were given?
5. What is the significance of rationalizing the denominator in trigonometric problems?

Tip: Always check for alternative trigonometric identities that might simplify complex expressions in word problems.