Question:
At a certain point in a large, level park, the angle of elevation to the top of an office building is ${30}^{\circ}$.
If you move ${400}ft$ closer to the building, the angle of elevation is ${45}^{\circ}$.
To the nearest ${10}ft$. how tall is the building?
My Initial Calculation :
For ${30}^{\circ}$ Triangle: $$ \begin{array}{1} =>\tan 30^{\circ}=\frac{h}{x+400}\\[0.1in] =>h=0.577(x+400)\\[0.1in] =>h=0.577x+230.8\\[0.1in] \end{array} $$
For ${45}^{\circ}$ Triangle: $$ \begin{array}{l} =>\tan 45^{\circ} = \frac{h}{x-400}\\[0.1in] =>\tan 45^{\circ} = \frac{0.577(x+400)}{x-400}\\[0.1in] =>\tan 45^{\circ} = \frac{0.577x+230.8}{x-400}\\[0.1in] =>x-400=0.577x+230.8\\[0.1in] =>x-0.577x=230.8+400\\[0.1in] =>0.423x=630.8\\[0.1in] =>x=\frac{630.8}{0.423}\\[0.1in] =>x=1491.25\\[0.1in] \end{array} $$
How tall is Building: $$ \begin{array}{1} =>\tan 30^{\circ}=\frac{h}{x+400}\\[0.1in] =>\tan 30^{\circ}=\frac{h}{1491.25+400}\\[0.1in] =>h=1091.91ft\\ \end{array} $$
The answer in the book is ${550ft}$, What wrong did I do in my calculation?
New Calculation :
For ${30}^{\circ}$ Triangle: $$ \begin{array}{1} =>\tan 30^{\circ}=\frac{h}{x+400}\\[0.1in] =>h=0.577(x+400)\\[0.1in] =>h=0.577x+230.8\\[0.1in] \end{array} $$
For ${45}^{\circ}$ Triangle: $$ \begin{array}{l} =>\tan 45^{\circ} = \frac{h}{x}\\[0.1in] =>\tan 45^{\circ} = \frac{0.577(x+400)}{x}\because h=0.577(x+400)\\[0.1in] =>\tan 45^{\circ} = \frac{0.577x+230.8}{x}\\[0.1in] =>x=0.577x+230.8\\[0.1in] =>x-0.577x=230.8\\[0.1in] =>0.423x=230.8\\[0.1in] =>x=\frac{230.8}{0.423}\\[0.1in] =>x=545.63\\[0.1in] \end{array} $$
How tall is Building: $$ \begin{array}{1} =>\tan 30^{\circ}=\frac{h}{x+400}\\[0.1in] =>\tan 30^{\circ}=\frac{h}{545.63+400}\because x=545.63\\[0.1in] =>h=545.95ft\\ \text{Rounded to the nearest 10}ft\\ =>h=550ft \end{array} $$
Pictorial View , with 3 variations , all will give the Same Solution :
In Blue Case , we have :
$\tan{30}=h/(x+400)$
$\tan{45}=h/(x)$
where $x$ is the Base length of the Smaller triangle.
In Green Case , we have :
$\tan{30}=h/(y)$
$\tan{45}=h/(y-400)$
where $y$ is the Base length of the Larger triangle.
In Purple Case , we have :
$\tan{30}=h/(z+200)$
$\tan{45}=h/(z-200)$
where $z$ is the Mid-Point of the Line Segment between the Bases of the 2 triangles.
All variations will give the Same Equivalent Solution.
Where OP went wrong :
What OP did was mostly Correct in trigonometric terms.
Where OP went wrong is : Mixing the Bases.
Like the Purple Case , OP took $x+400$ & $x-400$ , which is not the Criteria given & that will give wrong ( Double ! ) height.
Height will be scaled up & that will not match the text book.