We need to find the height of the tree. It is represented by x in the following triangle:
Using the Pythagorean Theorem, we have:
[tex]\begin{gathered} (1+3x)^{2}=x^{2}+35^{2} \\ \\ 1+6x+9x^{2}=x^{2}+1225 \\ \\ 1+6x+9x^{2}-x^{2}-1225=0 \\ \\ 8x^{2}+6x-1224=0 \end{gathered}[/tex]
Now, using the quadratic formula, we obtain:
[tex]\begin{gathered} x=\frac{-6\pm\sqrt[]{6^{2}-4(8)(-1224)}}{2(8)} \\ \\ x=\frac{-6\pm\sqrt[]{36+39.168}}{16} \\ \\ x=\frac{-6\pm\sqrt[]{36+39.168}}{16} \\ \\ x=\frac{-6\pm\sqrt[]{39204}}{16} \\ \\ x=\frac{-6\pm198}{16} \\ \\ x_1=\frac{-6-198}{16}=\frac{204}{16}=12.75 \\ \\ x_2=\frac{-6+198}{16}=\frac{192}{6}=12 \end{gathered}[/tex]
Since x is the height of the tree, it needs to be a positive value. Then, only x₂ is possible.
Therefore, the height of the tree is 12 ft.
