Question

The faces of the president at Mount Rushmore are 60 feet tall. A visitor see the top of George Washington's head at a 48° angle of elevation and his chin at a 44.76° angle of elevation. Find the height of Mount Rushmore.

Answer 1

First, we can make a drawing to better understand the information given:

We need to find the height of Mount Rushmore, represented by x in the drawing.

We can identify that we have a right triangle, therefore we can use the trigonometry relationships of the right triangle.

We have from the small triangle,

`\begin{gathered} \tan \theta=\frac{opposite\text{ side}}{\text{adjacent side}} \\ \tan (44.76)=(x)/(z) \\ z=(x)/(\tan (44.76)) \end{gathered}`

And from the big triangle we have,

`\begin{gathered} \tan (48)=((60+x))/(z) \\ z=((60+x))/(\tan (48)) \end{gathered}`

Now we can equate both equations and solve for x:

`\begin{gathered} (x)/(\tan(44.76))=((60+x))/(\tan (48)) \\ x\tan (48)=\tan (44.76)\cdot(60+x) \\ x\tan (48)=60\tan (44.76)+x\tan (44.76) \\ x\lbrack\tan (48)-\tan (44.76)\rbrack=60\tan (44.76) \\ x=(60\cdot\tan (44.76))/(\lbrack\tan (48)-\tan (44.76)\rbrack)=500.18 \end{gathered}`

Answer: The height of Mount Rushmore is about 500 ft.