Given:
Length of the ladder = 9.85 m
The height to the top of the ladder is 5 m more than the distance between the wall and the foot of the ladder.
To find:
The height to the top and the distance between the wall and the foot of the ladder.
Solution:
let x be the distance between the wall and the foot of the ladder. Then the height to the top of the ladder is (x+5).
Pythagoras theorem: In a right angle triangle,
[tex]Hypotenuse^2=Base^2+Perpendicular^2[/tex]
In the given situation, hypotenuse is the length of ladder, i.e., 9.85 m. The base is x m and the height is (x+5) m.
Using the Pythagoras theorem, we get
[tex](9.85)^2=x^2+(x+5)^2[/tex]
[tex]97.0225=x^2+x^2+10x+25[/tex]
[tex]0=2x^2+10x+25-97.0225[/tex]
[tex]0=2x^2+10x-72.0225[/tex]
Here, [tex]a=2, b=10,c=-72.0225[/tex]. Using the quadratic formula, we get
[tex]x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
[tex]x=\dfrac{-10\pm \sqrt{(10)^2-4(2)(-72.0225)}}{2(2)}[/tex]
[tex]x=\dfrac{-10\pm \sqrt{676.18}}{4}[/tex]
Approximating the value, we get
[tex]x=\dfrac{-10\pm 26}{4}[/tex]
[tex]x=\dfrac{-10+26}{4},\dfrac{-10-26}{4}[/tex]
[tex]x=\dfrac{16}{4},\dfrac{-36}{4}[/tex]
[tex]x=4,-9[/tex]
Distance cannot be negative so [tex]x\neq -9.[/tex]
Now we have [tex]x=4[/tex]
[tex]x+5=4+5[/tex]
[tex]x+5=9[/tex]
Therefore, the base is 4 m and the height is 9 m.
