Mathematics: Upper AA 66-ft-tall fence runs parallel to a wall of a house at a distance of 3030 ft. Find the length of the shortest ladder that extends from the ground, over the fence, to the house. Assume the vertical wall of the house and the horizontal ground have infinite extent.

she would have to work 8 hours...

Upper AA 66-ft-tall fence runs parallel to a

mattybacchetti

22.01.2022

0.33 ft

Step-by-step explanation:

From the diagram, triangle ACE is similar to triangle BCD.

Therefore,

$\frac{b}{30+b} = \frac{6}{a}$

$a=\frac{6(30+b)}{b}$

Let the Length of the Ladder, L

Using Pythagoras theorem

$L^{2}=(30+b)^{2}+(\frac{6(30+b)}{b})^{2}$

$L^{2}=\frac{(30+b)^{2}(36+b^{2})}{b^{2}}$

$L=\sqrt{\frac{(30+b)^{2}(36+b^{2})}{b^{2}} }$

$L=\frac{30+b}{b}\sqrt{b^2+36}$

To find the least value, we have to equate the derivative of L to 0.

$L^{1}=\frac{b^{3}-1080}{b^{2}\sqrt{b^2+36}}$

Therefore:

b³-1080=0

b³=1080

b=10.26

The minimum length of the ladder is:

$L=\frac{30+10.26}{10.26}\sqrt{10.26^2+36}$

=0.33 ft

0,0(0 оценок)