fence is 1.5m high and 1m from a wall.ladder must start from ground, touch top of fence, and rest somewhere on wall. Calculate the minimum length of ladder.

Question

The eq'n for length of a ladder i came up with is:

L = 1.5/sinx + 1/cosx

L'= 1.5cosx/sin^2x - (-sinx)/cos^2x

and then set that to zero. 

i couldn't solve for x and if i did i don't think the answer would be correct.  

 

How do i answer this using calculus?

Answer 1

Hi Madison,

 

You are on the right track and your work so far is correct! To solve L' = 0 and figure out the minimum of L:

 

I found it easier to think of L this way for the purpose of taking the derivative:

 

L = 1.5csc(x) + sec(x)

L' = -1.5csc(x)cot(x) + sec(x)tan(x)

L' = -1.5cos(x)/sin(x)^2 + sin(x)/cos(x)^2 = 0

 

1.5cos(x)/sin(x)^2 = sin(x)/cos(x)^2    next multiply by the denominators:

1.5cos(x)^3 = sin(x)^3

1.5 = tan(x)^3

x = arctan(1.5^(1/3))

 

x ~ 0.8528 rad will give you the proper angle of the ladder, simply plug it into L to get L min ~ 3.5117m

I  verified this using a graph, should be right

 

Cheers!