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
3.51 is correct
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
To find the minimum length of the ladder, you can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, you have a right triangle with a height of 1.5m (opposite side) and a base of 1m (adjacent side). Let's solve for the length of the hypotenuse (the ladder).
Using the Pythagorean theorem equation:
hypotenuse^2 = height^2 + base^2
Let's substitute the given values into the equation:
hypotenuse^2 = 1.5^2 + 1^2
hypotenuse^2 = 2.25 + 1
hypotenuse^2 = 3.25
Now, to find the length of the hypotenuse (the ladder), you need to take the square root of both sides:
hypotenuse = sqrt(3.25)
Calculating the square root of 3.25, we get approximately 1.8, not 4.5 as the answer in the book suggests.
It seems that there may be an error in the answer key or a mistake in the question itself. Double-check your calculations and make sure you have correctly entered the values before assuming there is an error in the book.