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:
The distance along the ladder to the top of the fence is=?
Write a function L(x) which gives the total length of a ladder which touches the ground at an angle x, touches the top of the fence and just reaches the wall=?
The length of the shortest ladder is=?
If the foot of the ladder is d feet from the fence,
3/d = tan x
d = 3cot x
L(x) = (d+3) sec x
= 3(1+cotx)secx
dL/dx = 3secx(tanx - csc^2(x)+1)
So, either secx=0 (no solution), or
tanx - csc^2(x)+1 = 0
x = π/4
So, L(π/4) = 3(2)(√2) = 6√2
As expected, a square has the shortest diagonal.
Correction: If the ladder makes an angle 1.25 radians with the ground, touches the top of the fence, and just reaches the wall, the distance along the ladder from the ground to the top of the fence=?
The distance along the ladder from the top of the fence to the wall is=?
come on, man.
distance to top of fence:
d/3 = csc(1.25)
You have the formula for L(x), the length of the ladder, so subtract d from that.
To find the distance along the ladder to the top of the fence, we can use the Pythagorean theorem. According to the theorem, the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
In this case, the hypotenuse of the right triangle is the length of the ladder, the base is the distance from the fence to the building, and the height is the height of the fence.
Let's call the distance along the ladder to the top of the fence as 'd'.
Using the Pythagorean theorem, we can write the equation as:
d^2 = 3^2 + 3^2
d^2 = 9 + 9
d^2 = 18
Taking the square root of both sides, we get:
d = √18
Therefore, the distance along the ladder to the top of the fence is approximately 4.2426 feet.
Now, let's define the function L(x) to find the total length of the ladder which touches the ground at an angle x, touches the top of the fence, and just reaches the wall.
In this case, we can consider the ladder as the hypotenuse of a right triangle. The base of the triangle would be the distance from the fence to the building (3 feet), and the height would be the height of the fence (3 feet).
Using the trigonometric relationship of sine, the length of the ladder can be calculated as:
L(x) = [3 / sin(x)]
Finally, to find the length of the shortest ladder, we need to find the minimum value of L(x). This can be done by finding the minimum value of sin(x), which occurs when x = 90 degrees (or π/2 radians).
Therefore, the length of the shortest ladder is:
L(90 degrees) = [3 / sin(90 degrees)]
L(90 degrees) = 3 / 1
L(90 degrees) = 3 feet.