Answer:
0.33 ft
Step-by-step explanation:
From the diagram, triangle ACE is similar to triangle BCD.
Therefore,
[TeX]\frac{b}{30+b} = \frac{6}{a}[/TeX]
[TeX]a=\frac{6(30+b)}{b}[/TeX]
Let the Length of the Ladder, L
Using Pythagoras theorem
[TeX] L^{2}=(30+b)^{2}+(\frac{6(30+b)}{b})^{2} [/TeX]
[TeX] L^{2}=\frac{(30+b)^{2}(36+b^{2})}{b^{2}}[/TeX]
[TeX] L=\sqrt{\frac{(30+b)^{2}(36+b^{2})}{b^{2}} }[/TeX]
[TeX] L=\frac{30+b}{b}\sqrt{b^2+36}[/TeX]
To find the least value, we have to equate the derivative of L to 0.
[TeX] L^{1}=\frac{b^{3}-1080}{b^{2}\sqrt{b^2+36}}[/TeX]
Therefore:
b³-1080=0
b³=1080
b=10.26
The minimum length of the ladder is:
[TeX] L=\frac{30+10.26}{10.26}\sqrt{10.26^2+36}[/TeX]
=0.33 ft
