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,
[tex]\begin{gathered} \tan \theta=\frac{opposite\text{ side}}{\text{adjacent side}} \\ \tan (44.76)=\frac{x}{z} \\ z=\frac{x}{\tan (44.76)} \end{gathered}[/tex]And from the big triangle we have,
[tex]\begin{gathered} \tan (48)=\frac{(60+x)}{z} \\ z=\frac{(60+x)}{\tan (48)} \end{gathered}[/tex]Now we can equate both equations and solve for x:
[tex]\begin{gathered} \frac{x}{\tan(44.76)}=\frac{(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=\frac{60\cdot\tan (44.76)}{\lbrack\tan (48)-\tan (44.76)\rbrack}=500.18 \end{gathered}[/tex]Answer: The height of Mount Rushmore is about 500 ft.
