A fence 5 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?
draw a diagram. If the foot of the ladder is x feet from the fence, and the ladder reaches y feet up the wall, then we have
x/5 = (x+3)/y
y = 5 + 15/x
if the length of the ladder is z,
z^2 = (x+3)^2 + y^2
= (x+3)^2 + (5 + 15/x)^2
z = √((x+3)^2 + (5 + 15/x)^2)
dz/dx = (x+3)(x^3-75) / <a bunch of nonzero junk>
so, dz/dx=0 at x = -3 or ∛75
z^2 = (3+∛75)^2 + (5+15/∛75)^2
z = 11.19 feet
Ah, the shortest ladder dilemma! Well, let me bring out the clown calculator and see what we've got here.
So, we have a fencing situation, a 5-foot-tall fence that stands parallel to a tall building. And it's cheekily positioned 3 feet away from the building, just to make things interesting.
To determine the length of the shortest ladder that can reach from the ground over the fence to the building, we can use a delightful little geometric trick called the Pythagorean Theorem.
In plain English, we have a right triangle, where the height of the fence is one leg, the distance from the fence to the building is the other leg, and the ladder is the hypotenuse. Let's do some clown math!
According to the Pythagorean Theorem, the square of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides. So, we have:
Ladder² = Fence Height² + Distance to the Building²
Plugging in the numbers, we get:
Ladder² = 5² + 3²
Ladder² = 25 + 9
Ladder² = 34
Now, square root both sides to find the length of the ladder:
Ladder = √34
And there you have it! The shortest ladder that will reach from the ground over the fence to the building is approximately √34 feet long.
Just remember, no matter the length of the ladder, always use caution when climbing and make sure the clowns are fully trained in ladder acrobatics! Safety first!
To find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building, let's use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled 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, the ladder will act as the hypotenuse, the fence will be one side, and the distance between the fence and the building will be the other side.
Let's denote the length of the ladder as "x".
According to the Pythagorean theorem, we can write the equation as:
x^2 = 3^2 + 5^2
Simplifying:
x^2 = 9 + 25
x^2 = 34
Taking the square root of both sides:
x = √34
So, the length of the shortest ladder that will reach from the ground over the fence to the wall of the building is approximately √34 feet.
To determine 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 a right-angled 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 scenario, the ladder acts as the hypotenuse, the height of the fence acts as one side, and the distance between the fence and the building acts as the other side.
Let's represent the length of the ladder as 'L,' the height of the fence as 'h,' and the distance between the fence and the building as 'd.'
Using the Pythagorean theorem, we have the following equation:
L^2 = h^2 + d^2
Substituting the given measurements, we have:
L^2 = 5^2 + 3^2
Simplifying the equation:
L^2 = 25 + 9
L^2 = 34
To find the length of the ladder, we need to take the square root of both sides:
L = √34
So, the length of the shortest ladder that will reach from the ground over the fence to the wall of the building is approximately equal to √34 feet.