Posted by lala on Saturday, April 10, 2010 at 11:43pm.
A fence is 1.5m high and is 1m from a wall. A ladder must start from the ground, touch the top of the fence, and rest somewhere on the wall. Calculate the minimun lenght of the ladder.
I drew a diagram and i ended up with a right angle triangle. i made the hypotenuse the lenght of the ladder, h=1.5 and base=1. I then used the pytagorean theorem to solve for the hypotenuse, but my answer doesnt match the one at the back of the book(4.5m)
Calculus - drwls, Sunday, April 11, 2010 at 7:58am
Let A be the angle that the ladder makes with the horizontal. The ladder's length L must be such that
1.5/sinA + 1/cos A = L
Compute dL/dA and set it equal to zero to get the angle A for which the length is minimum.
dL/dA = -1.5 cosA/sin^2A + sinA/cos^2A = 0
1.5 cos^3A = sin^3A
tan^3A = 1.5
tanA = 1.145
A = 48.86 degrees
L = 1.5/sin48.86 + 1/cos48.86
= 1.992 + 1.520 = 3.512
I don't agree with your book's answer, either.
Concur with dwrls, book is wrong - Calculus - Reiny, Sunday, April 11, 2010 at 9:31am
I tried it all algebraically, no trig.
let the ladder reach y m up the wall, and touch the ground x m from the fence.
So I had two similar right angled triangles and
1.5/x = y /(1+x)
xy = 1.5 + 1.5x
2xy = 3 + 3x
y = (3+3x)/(2x)
L^2 = y^2 + )1+x)^2
= [(3+3x)/(2x)]^2 + (1+x)^2
= (9 + 18x + 9x^2)/(4x^2) + 1 + 2x + x^2
=(9/4)x^-2 + (9/2)x^-1 + 9/4 + 1 + 2z + x^2
2L(dL/dx) = (-18/4)x^-3 - (9/2)x^-2 + 2 + 2x
= 0 for a max/min of L
(-18/4)x^-3 - (9/2)x^-2 + 2 + 2x
-18 - 18x + 8x^3 + 8x^4 = 0
8x^3(1+x) - 18(1+x) = 0
(1+x)(8x^3 - 18) = 0
x = -1, silly answer or
x^3 = 18/8
x = cuberoot(18)/2 = 1.31037
sub that back into y = ..
y = 2.6447
then L^2 = y^2 + (1+x)^2
L = 3.5117
Calculus - Anonymous, Monday, April 15, 2013 at 5:27pm
dwls solution is corret but when you plug 48.86 degrees back into the original equation your calculator must be in rads and you will get 4.5m
Answer This Question
More Related Questions
- Calculus - A fence 5 feet tall runs parallel to a tall building at a distance of...
- Math - A ladder is placed against a wall, with its base 2 m from the wall. The ...
- Calculus - A fence 3 feet tall runs parallel to a tall building at a distance of...
- Mathmatics 10 - A ladder is placed so its foot is 2m from a wall. The ladder ...
- Calculus - A 17 foot ladder is leaning on a 6 foot fence. The base of the ladder...
- Calculus - A 25 foot ladder is leaning on a 9 foot fence. The base of the ladder...
- Calculus - 1a. 15 ft ladder is placed against a vertical wall. the bottom of the...
- Calculus - A 23 foot ladder is leaning on a 6 foot fence. The base of the ladder...
- Trig - The foot of a ladder is on level ground 1.5m from a wall. The ladder ...
- CALCULUS - A 25 ft ladder is leaning against a vertical wall. At what rate (with...