The angle of elevation of the top of a tree is found to be 23° at one point and 76° at a point 49 feet nearer the tree. How high is the tree if both observation points and the base of the tree are on the same horizontal plane?
+26387 The angle of elevation of the top of a tree is found to be 23° at one point and 76° at a point 49 feet nearer the tree. How high is the tree if both observation points and the base of the tree are on the same horizontal plane?
\(\begin{array}{rcll} \tan{(23^\circ)} &=& \frac{h}{x} \\ x &=& \frac{h}{ \tan{(23^\circ)}} \\\\ \tan{(76^\circ)} &=& \frac{h}{x-49} \\ \tan{(76^\circ)} &=& \frac{h}{\frac{h}{ \tan{(23^\circ)}}-49} \\ \left( \frac{h}{ \tan{(23^\circ)}}-49 \right)\cdot \tan{(76^\circ)} &=& h\\ h\cdot \frac{\tan{(76^\circ)}} { \tan{(23^\circ)}}-49\cdot \tan{(76^\circ)}&=& h\\ h\cdot \left( \frac{\tan{(76^\circ)}} { \tan{(23^\circ)}} - 1 \right) &=& 49\cdot \tan{(76^\circ)}\\ h\cdot \left( \frac{\tan{(76^\circ)}-\tan{(23^\circ)} } { \tan{(23^\circ)}} \right) &=& 49\cdot \tan{(76^\circ)}\\ h &=& 49\cdot \left( \frac{ \tan{(76^\circ)} \cdot \tan{(23^\circ)} } { \tan{(76^\circ)}-\tan{(23^\circ)} } \right) \\ h &=& 49\cdot \left( \frac{4.01078093354 \cdot 0.42447481621 } { 4.01078093354-0.42447481621 } \right) \\ h &=& 49\cdot \left( \frac{1.70247549962 } { 3.58630611733 } \right) \\ h &=& 49\cdot 0.47471561097 \\ h &=& 23.2610649376\ \text{feet} \\ \end{array}\)
The tree is 23.2610649376 feet high
+26387 The angle of elevation of the top of a tree is found to be 23° at one point and 76° at a point 49 feet nearer the tree. How high is the tree if both observation points and the base of the tree are on the same horizontal plane?
\(\begin{array}{rcll} \tan{(23^\circ)} &=& \frac{h}{x} \\ x &=& \frac{h}{ \tan{(23^\circ)}} \\\\ \tan{(76^\circ)} &=& \frac{h}{x-49} \\ \tan{(76^\circ)} &=& \frac{h}{\frac{h}{ \tan{(23^\circ)}}-49} \\ \left( \frac{h}{ \tan{(23^\circ)}}-49 \right)\cdot \tan{(76^\circ)} &=& h\\ h\cdot \frac{\tan{(76^\circ)}} { \tan{(23^\circ)}}-49\cdot \tan{(76^\circ)}&=& h\\ h\cdot \left( \frac{\tan{(76^\circ)}} { \tan{(23^\circ)}} - 1 \right) &=& 49\cdot \tan{(76^\circ)}\\ h\cdot \left( \frac{\tan{(76^\circ)}-\tan{(23^\circ)} } { \tan{(23^\circ)}} \right) &=& 49\cdot \tan{(76^\circ)}\\ h &=& 49\cdot \left( \frac{ \tan{(76^\circ)} \cdot \tan{(23^\circ)} } { \tan{(76^\circ)}-\tan{(23^\circ)} } \right) \\ h &=& 49\cdot \left( \frac{4.01078093354 \cdot 0.42447481621 } { 4.01078093354-0.42447481621 } \right) \\ h &=& 49\cdot \left( \frac{1.70247549962 } { 3.58630611733 } \right) \\ h &=& 49\cdot 0.47471561097 \\ h &=& 23.2610649376\ \text{feet} \\ \end{array}\)
The tree is 23.2610649376 feet high
