To find the height of the tree given the change in the angle of elevation, let's denote:
- [tex]\( h \)[/tex] as the height of the tree,
- [tex]\( x \)[/tex] as the initial distance from the observer to the base of the tree.
The key facts are:
1. The angle of elevation changes from [tex]\(45^{\circ}\)[/tex] to [tex]\(30^{\circ}\)[/tex] when the observer moves 5 meters closer to the tree.
We will use trigonometric relationships for both positions of the observer:
### Step 1: Setting up the equations
Initially, the angle of elevation is [tex]\(45^{\circ}\)[/tex] when the observer is [tex]\( x \)[/tex] meters away from the tree. Using the tangent function, we have:
[tex]\[ \tan(45^{\circ}) = \frac{h}{x} \][/tex]
Since [tex]\( \tan(45^{\circ}) = 1 \)[/tex]:
[tex]\[ 1 = \frac{h}{x} \][/tex]
Thus,
[tex]\[ h = x \][/tex]
When the observer moves 5 meters closer, the angle of elevation changes to [tex]\(30^{\circ}\)[/tex]. Now, the observer is [tex]\( x - 5 \)[/tex] meters away from the base of the tree. Again using the tangent function:
[tex]\[ \tan(30^{\circ}) = \frac{h}{x - 5} \][/tex]
Since [tex]\(\tan(30^{\circ}) = \frac{1}{\sqrt{3}}\)[/tex]:
[tex]\[ \frac{1}{\sqrt{3}} = \frac{h}{x - 5} \][/tex]
So,
[tex]\[ h = \frac{x - 5}{\sqrt{3}} \][/tex]
### Step 2: Solving the system of equations
We now have two equations:
1. [tex]\( h = x \)[/tex]
2. [tex]\( h = \frac{x - 5}{\sqrt{3}} \)[/tex]
Substituting [tex]\( h = x \)[/tex] from the first equation into the second equation, we get:
[tex]\[ x = \frac{x - 5}{\sqrt{3}} \][/tex]
Multiply both sides by [tex]\( \sqrt{3} \)[/tex]:
[tex]\[ x\sqrt{3} = x - 5 \][/tex]
Rearrange to solve for [tex]\( x \)[/tex]:
[tex]\[ x\sqrt{3} - x = -5 \][/tex]
[tex]\[ x(\sqrt{3} - 1) = 5 \][/tex]
[tex]\[ x = \frac{5}{\sqrt{3} - 1} \][/tex]
To simplify [tex]\( x \)[/tex], multiply numerator and denominator by the conjugate [tex]\( \sqrt{3} + 1 \)[/tex]:
[tex]\[ x = \frac{5(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} \][/tex]
[tex]\[ x = \frac{5(\sqrt{3} + 1)}{3 - 1} \][/tex]
[tex]\[ x = \frac{5(\sqrt{3} + 1)}{2} \][/tex]
Now, substituting [tex]\( x \)[/tex] back into the height equation [tex]\( h = x \)[/tex]:
[tex]\[ h = \frac{5(\sqrt{3} + 1)}{2} \][/tex]
Calculating the numerical value:
[tex]\[ h \approx 6.83 \, \text{meters} \][/tex]
Hence, the height of the tree is approximately [tex]\( 6.83 \)[/tex] meters.
According to the result, none of the provided answer options in the original problem statement matches exactly the simplified height. However, given the calculations, we can conclude that Option A. [tex]\( \frac{5 \sqrt{3}}{3 - \sqrt{3}} \)[/tex] implicitly represents the right approach for the problem that pairs with the calculated numeric conversion.
