A fence 3 feet tall runs parallel to a tall building at a distance of 6 feet from the building. What is the le

Question

A fence 3 feet tall runs parallel to a tall building at a distance of 6 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

Answer 1

Good answers!

BUT... (now here's a good one for your teacher / prof.. assuming your in school)

you assume the ladder nor the building has any velocity such that an observed length contraction would occur due to special relativity.

AND...

Where do you buy a 12.x*x foot ladder.. Most adjustable ladders aren't that long. The longest I could find went to 12 feet flat. You would need a 13 foot ladder.

Don't get on me for not answering specifically to the exact decimal.. just thought being a smart a$$ would be fun (for you and your professor). enjoy.

Answer 2

I set up this question in excel. I suggest you try it. We can assume that neither 0 degrees or 90 degrees will work, thus our range of angles will be 1 - 89 degrees.

I wish I knew how to put a drawing into this program, but let's call the distance from the bottom of the ladder to the fence x. That makes the distance from the ladder to the wall x+6. The angle is Theta. L is the length of the ladder.

My columns looked like this:

Column A has numbers 1 - 89

Column B has this formula: =3/(TAN(A2*PI()/180))

That solves for x (distance from bottom of ladder to fence)

Columb C has this formula: =B2+6

This solves for x+6 (distance from bottom of ladder to wall)

Column D has this forumula: =C2/(COS(A2*PI()/180))

This solves for L (length of ladder and also your answer)

For excel, you have to change the formula for trigonometric functions because it gives the answers in radians. So the formula you use is actually: =3/(TAN(A2*PI()/180))

The 3 is the height of the fence.

Then I just looked for the angle with the smallest L, which was in fact 38 degrees, not 45. (Initially, I thought 45 would be correct as well, but it's always better to do your own work).

at 38 degrees, the ladder will in fact be 12 feet, 5 inches, and 27/32nd's of an inch. If the question speficies to round to the nearest inch, it's 12 feet, 6 inches. It depends what is required from the question.

Hope this helps!

JRG

Answer 3

Ok, here's how you do it

let's say x is the distance between the base of the ladder and the fence. Y will be the height between the base of the building and the end of tha ladder that touches the wall. q will be the angle between the ground and the ladder and l will be the length of the ladder

You know that

tan q = 3ft/x

sin q = y/l

and

l^2=y^2+(6ft+x)^2

solving for x and y and substitutting

l^2 =l^2sin^2q +(6ft+3ft/tan q)^2

solving for l^2

l^2 = (6ft+3ft/tan q)^2/(1-sin^2q)

you know that 1-sin^2q = cos^2 q so

l^2 = (6ft+3ft/tan q)^2/cos^2q

squared root

l = (6ft+3ft/tan q)/cos q

doing a little math

l = 6ft/cos q +3ft/sin q

deriving for q

dl = (6ft*tan q/cos q - 3ft*cos q/ sin^2 q) dq

now in order to get the minimum size of the sair, you have to equal this equation to 0 and solve for q

you get that

q = 38.44º

and substituting in the original equation before deriving.

l min = 6ft/cos 38.44º + 3ft/sin 38.44º

lmin = 12.49 ft

hope this helps.