fence 3 feet tall runs parallel to a tall building at a distance of 7 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?
If the foot of the ladder is x feet from the fence, and it reaches to a height h on the wall, then using similar triangles,
x/3 = (x+7)/h
h = 3(x+7)/x
the ladder's length z is
z^2 = (x+7)^2 + h^2
= (x+7)^2 + (3(x+7)/x)^2
z = (x+7)√(x^2+9) / x
So, for minimum z, we need dz/dx = 0
dz/dx = (x^3-63) / x^2√(x^2+9)
dz/dx=0 when x=∛63, or about 4 feet
So, minimum z is roughly
(4+7)√(16+9)/4 = 55/4 = 13.75 ft
please i need help, oi was looking towards similar problem, but i'm not sure how cube route 75 came into play with one of the problems that had 5 feet tall running parallel at 3 ft distance
oh thank you. this formate shows a lot better. Thank you Steve.
in that case,
z^2 = (x+3)^2 + (5(x+3)/x)^2
z = (x+3)√(x^2+25) / x
dz/dx = (x^3-75) / x^2√(x^2+25)
and x = ∛75
To find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building, we can use the Pythagorean theorem.
The Pythagorean theorem states that in any right triangle, the square of the length 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, we can consider the ladder as the hypotenuse of a right triangle, the fence as one of the legs, and the distance from the fence to the building as the other leg.
Let's denote the length of the ladder as "L", the height of the fence as "h" (which is 3 feet in this case), and the distance from the fence to the building as "d" (which is 7 feet).
Applying the Pythagorean theorem, we have:
L^2 = h^2 + d^2
L^2 = 3^2 + 7^2
L^2 = 9 + 49
L^2 = 58
To find the length of the ladder, we take the square root of both sides:
L = √58
Using a calculator, we find that the square root of 58 is approximately 7.62.
Therefore, the length of the shortest ladder that will reach from the ground over the fence to the wall of the building is approximately 7.62 feet.