Question

A fox and an eagle lived at the top of a cliff of height \[6\,m\], whose base was at a distance of \[10\,m\] from a point A on the ground. The fox descended the cliff and went straight to point A. The eagle flew vertically up to a height \[x\] metres and then flew in a straight line to a point A, the distance travelled by each being the same. Find the value of \[x\].

Answer 1

Hint: The question is in the form of words. So we have to just go through the question and write the given data in the form of numbers or equation and then using the mathematical concepts we are going to solve the given question and determine the value of the unknown.

Complete step-by-step answer:

Now consider the given question.
In the question they have mentioned the word height and base this tells it follows the triangle plane figure.
By considering the word problem we draw a figure related to the question. The line represents the initial level. Hence the value of \[AB = 10\,m\] and \[BC = 6\, + x\,\,m\].
The distance travelled by fox is \[16\,m\]
The distance travelled by an eagle is AC and x . So we determine the value of AC. By using the Pythagoras theorem we have \[AC = \sqrt {{{(6 + x)}^2} + {{10}^2}} \]
The distance travelled by an eagle is given by \[AC + x\]. In the question they have mentioned the distance travelled by each being the same. So now we have
Distance travelled by fox = distance travelled by eagle.
\[ \Rightarrow 16 = AC + x\]
Taking x to LHS we have
\[ \Rightarrow 16 - x = AC\]
Squaring on both sides we have
\[ \Rightarrow {\left( {16 - x} \right)^2} = A{C^2}\]
On simplifying we have
\[ \Rightarrow 256 + {x^2} - 32x = 36 + {x^2} + 12x + 100\]
On further simplifying we have
\[ \Rightarrow 120 = 44x\]
On dividing by 44 we have
\[ \Rightarrow x = 2.73\,\,m\]
Hence we have determined the value of x.
So, the correct answer is “x = 2.73 m”.

Note: Here in this question we can easily guess the plane figure because they have already mentioned the word base and height. In the plane figure of the triangle only we have the word base and height. Suppose if there are other dimensions the plane will be different. For any mathematical problems the basic arithmetic operations are needed.