Calculus
posted by lala .
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)

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. 
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
= 0
times 4x^3
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
gave me
L = 3.5117 
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
Respond to this Question
Similar Questions

Calculus
A 25 foot ladder is leaning on a 9 foot fence. The base of the ladder is being pulled away from the fence at the rate of 10 feet/minute. How fast is the top of the ladder approaching the ground when the base is 9 from the fence? 
Calculus
A 17 foot ladder is leaning on a 6 foot fence. The base of the ladder is being pulled away from the fence at the rate of 5 feet/minute. How fast is the top of the ladder approaching the ground when the base is 6 from the fence? 
Math
A ladder is placed against a wall, with its base 2 m from the wall. The ladder touches the top of a 2 m fence that is 1.5 m from the wall. How high up the wall does the ladder reach? 
Mathmatics 10
A ladder is placed so its foot is 2m from a wall. The ladder touches the top of a 2m fence that is 1.5m from the wall. How high up the wall does the ladder reach? 
Calculus
A 23 foot ladder is leaning on a 6 foot fence. The base of the ladder is being pulled away from the fence at the rate of 7 feet/minute. How fast is the top of the ladder approaching the ground when the base is 6 from the fence? 
CALCULUS
A 25 ft ladder is leaning against a vertical wall. At what rate (with respect to time) is the angle theta between the ground and the ladder changing, if the top of the ladder is sliding down the wall at the rate of r inches per second, … 
Calculus
1a. 15 ft ladder is placed against a vertical wall. the bottom of the ladder is sliding away from the wall at a rate of 2 ft/sec. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 9 ft from the … 
Calculus
A fence 3 feet tall runs parallel to a tall building at a distance of 3 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 the following: … 
Calculus
A fence 5 feet tall runs parallel to a tall building at a distance of 2 feet from the building. We want to find the the length of the shortest ladder that will reach from the ground over the fence to the wall of the building. Here … 
Trig
The foot of a ladder is on level ground 1.5m from a wall. The ladder leans agents the wall. The angle formed by the ladder and the ground is 70 degrees. Calculate how high up the wall the ladder reaches.