The height of the tree is 60 meters.
Explanation:
Let the height of the tree be x. The tree casts a shadow of [tex]x-49[/tex] meters and the distance from the top of the tree to the end of the shadow is [tex]x+1[/tex] meters.
The sides of the triangle are attached in the image below:
Using pythagoras theorem,
[tex]x^{2}+(x-49)^{2}=(x+1)^{2}[/tex]
Expanding, we get,
[tex]2 x^{2}-98 x+2401=x^{2}+2 x+1[/tex]
[tex]2 x^{2}-98 x+2400=x^{2}+2 x[/tex]
[tex]2 x^{2}-100 x+2400=x^{2}[/tex]
[tex]x^{2}-100 x+2400=0[/tex]
Solving the equation using the quadratic formula [tex]x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}[/tex], we get,
[tex]x=\frac{-(-100)\pm\sqrt{(-100)^{2}-4 \cdot 1 \cdot 2400}}{2 \cdot 1}[/tex]
Simplifying, we have,
[tex]x=\frac{100\pm\sqrt{10000-9600}}{2}[/tex]
[tex]x=\frac{100\pm\sqrt{400}}{2}[/tex]
[tex]x=\frac{100\pm{20}}{2}[/tex]
Thus,
[tex]x=\frac{100+20}{2} \\x=\frac{120}{2} \\x=60[/tex] and [tex]x=\frac{100-20}{2} \\x=\frac{80}{2} \\x=40[/tex]
where the value [tex]x=40[/tex] is not possible because substituting the value [tex]x=40[/tex] in [tex]x-49[/tex] results in negative solution. Which is not possible.
Hence, the value of x is 60.
Thus, The height of the tree is 60 meters.
