Question

At a certain time of day, a tree that is x meters tall casts a shadow that x - 17 meters long. If the distance from the top of the tree to the end of the shadow is x + 1 meters, what is the height, X_{1} of the tree?

Answer 1

Diagram:

From the above right triangle, we can stablish the following:

`\begin{gathered} pythagorean\text{ }theorem:a^2+b^2=c^2 \\ \left(x\right?^2+\left(x-17\right)^2=\left(x+1\right)^2 \end{gathered}`

Solving for x:

`\begin{gathered} x^2+x^2-34x+289=x^2+2x+1 \\ x^2+x^2-x^2-34x-2x+289-1=0 \\ x^2-36x+288=0 \end{gathered}`

apply the quadratic formula,

`\begin{gathered} x_(1,\:2)=(-b\pm √(b^2-4ac))/(2a) \\ x_(1,\:2)=(-\left(-36\right)\pm √(\left(-36\right)^2-4\cdot \:1\cdot \:288))/(2\cdot \:1) \end{gathered}`

then,

`x_(1,\:2)=(-\left(-36\right)\pm \:12)/(2\cdot \:1)`

separate the solutions,

`x_1=(-\left(-36\right)+12)/(2\cdot \:1),\:x_2=(-\left(-36\right)-12)/(2\cdot \:1)`

Thus,

`x=24,\:x=12`

since, the shadow is x-17 meters long, then any value below 17 does not make much sense, therefore,

The answer is, x = 24