From the problem we know that:
• x is the length of the ladder,
,
• the top of the ladder is 3 feet less than the length of the ladder,
,
• the distance from the bottom of the ladder to the building is 9 feet.
Adding the data of the problem to the drawing, we have:
We see that the ground, the wall and the ladder forms a right triangle of sides a, b and h, where:
• the hypotenuse is h = x,
,
• one cathetus is a = 9,
,
• the second cathetus is b = x - 3.
Pitagoras Theorem states that:
[tex]h^2=a^2+b^2.[/tex]
Replacing the values of a, b and h, we have the following equation for x:
[tex]x^2=9^2+(x-3)^2.[/tex]
We solve for x the equation:
[tex]\begin{gathered} x^2=81+x^2-2\cdot3\cdot x+3^2, \\ x^2=81+x^2-6x+9, \\ 0=90-6x, \\ 6x=90, \\ x=\frac{90}{6}=15. \end{gathered}[/tex]
Answer
The length of the ladder is x = 15 feet.
