Answer:
25.3 ft
Explanation:
The illustration of the problem is shown the attached image.
The length of the ladder can be calculated using the Pythagoras theorem:
[tex]Hypotenuse^2 = opposite^2 + adjacent^2[/tex]
The hypotenuse is the length of the ladder.
[tex]Hypotenuse = \sqrt{opposite^2 + adjacent^2}[/tex]
[tex]L^2 = BC^2 + (24 + AE)^2[/tex]........1
Triangle ABC is similar to triangle AEF, hence:
[tex]\frac{BC}{8} = \frac{AE + 24}{AE}[/tex]
BC = [tex]\frac{8(AE + 24)}{AE}[/tex].................2
Substitute 2 into 1
[tex]L^2[/tex] = [tex](\frac{8(AE + 24)}{AE})^2[/tex] + [tex](24 + AE)^2[/tex]
Let AE = x
[tex]L^2[/tex] = [tex](\frac{8x + 192}{x})^2 + (24 + x)^2[/tex]
= [tex](8 + \frac{192}{x})^2 + (24 + x)^2[/tex]
Minimize L with respect to x.
2[tex]\frac{dL}{dx}[/tex] = [tex]2(8 + \frac{192}{x})(-\frac{192}{x^2}) + 2(24 + x)[/tex]
=
