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scoundrel [369]
2 years ago
6

At a certain time of the day, a tree that is x meters tall casts a shadow that is x-49 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 of the tree ?

Mathematics
1 answer:
nikklg [1K]2 years ago
7 0

The height of the tree is 60 meters.

Explanation:

Let the height of the tree be x. The tree casts a shadow of x-49 meters and the distance from the top of the tree to the end of the shadow is x+1 meters.

The sides of the triangle are attached in the image below:

Using pythagoras theorem,

x^{2}+(x-49)^{2}=(x+1)^{2}

Expanding, we get,

2 x^{2}-98 x+2401=x^{2}+2 x+1

2 x^{2}-98 x+2400=x^{2}+2 x

2 x^{2}-100 x+2400=x^{2}

x^{2}-100 x+2400=0

Solving the equation using the quadratic formula x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}, we get,

x=\frac{-(-100)\pm\sqrt{(-100)^{2}-4 \cdot 1 \cdot 2400}}{2 \cdot 1}

Simplifying, we have,

x=\frac{100\pm\sqrt{10000-9600}}{2}

x=\frac{100\pm\sqrt{400}}{2}

x=\frac{100\pm{20}}{2}

Thus,

x=\frac{100+20}{2} \\x=\frac{120}{2} \\x=60  and  x=\frac{100-20}{2} \\x=\frac{80}{2} \\x=40

where the value x=40 is not possible because substituting the value x=40 in x-49 results in negative solution. Which is not possible.

Hence, the value of x is 60.

Thus, The height of the tree is 60 meters.

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