An 8 foot tall fence separates Larry's yard from Evan's yard.The fence is 3 feet from Larry's house and runs parallel to the side of his house. Larry wants to paint his house and needs to position a ladder extending from Evan's yard over the fence to his house. Assuming the vertical wall of Larry's house is less than 30 feet tall and Evan's yard is at least 20 feet long what is the length of the shortest ladder whose base is in Evan's yard, clears the fence and reaches Larry's house?

if h is the height of the ladder on the wall, and x is the distance of the base of the ladder from the wall, then by similar triangles,

x/h = (x-3)/8
h = 8x/(x-3)

Now, the ladder length y is

y^2 = x^2 + h^2 = x^2 + 64x^2/(x-3)^2
2yy' = 2x + 128x/(x-3)^2 - 128x^2/(x-3)^3

y' = (2x(x-3)^3 + 128x(x-3) - 128x^2)/y(x-3)^2

That's messy, but we want y'=0, and as long as the denominator isn't 0, we just need

2x(x-3)^3 + 128x(x-3) - 128x^2 = 0
x = 3 + 4∛3 = 8.769

h = 8x/(x-3) = 12.161

so, y, the ladder, is 15 ft

Well, it looks like Larry and Evan are in a bit of a height predicament, huh? They have an 8-foot tall fence standing between them and now they need a ladder to bridge that gap. Let's see if we can solve this.

Given that the fence is 8 feet tall, we need a ladder that extends over the fence and reaches Larry's house. Since Larry's house is less than 30 feet tall, we can assume the ladder needs to be around that length, give or take.

Now, the ladder needs to have its base in Evan's yard and clear the 8-foot fence. Therefore, the ladder should be at least 8 feet long to reach the top of the fence. But since we want the shortest ladder, we'll look for the most efficient solution.

Given that Evan's yard is at least 20 feet long and the ladder needs to clear an 8-foot fence, we have a right-angled triangle situation going on. The shorter leg of the triangle is 8 feet (the height of the fence), and the longer leg is at least 20 feet (the length of Evan's yard).

To find the length of the hypotenuse, which represents the ladder, we can use the Pythagorean theorem: a² + b² = c².

Plugging in the values, we have 8² + 20² = c². Solving it, we get 64 + 400 = c². That's 464 = c². Taking the square root of both sides, we find c ≈ 21.54 feet.

So, the length of the shortest ladder that clears the fence and reaches Larry's house is approximately 21.54 feet. Just remember, Larry and Evan, safety first! Watch your step when climbing over that fence!

To find the length of the shortest ladder that reaches Larry's house, we can use the Pythagorean theorem.

Let's label the ladder's length as "c", Larry's house height as "a", and the distance from the base of the ladder to the fence as "b".

According to the given information, the height of Larry's house is less than 30 feet, so we can assume "a" is less than 30.

We can create a right triangle, where the ladder is the hypotenuse, and the vertical side is a (Larry's house height), and the horizontal side is b (distance from the base of the ladder to the fence).

Using the Pythagorean theorem, we have the equation: a^2 + b^2 = c^2.

Given that a = 8 feet and b = 3 feet, we can substitute these values into the equation: (8^2) + (3^2) = c^2.

Simplifying, we get: 64 + 9 = c^2.

Combining, we have: 73 = c^2.

To find c, we can take the square root of both sides: √73 = √(c^2).

Therefore, the length of the shortest ladder that reaches Larry's house is approximately 8.54 feet.

To find the length of the shortest ladder, we can use the Pythagorean theorem.

Let's label the length of the ladder as "c", the height of the fence as "a," and the distance from the fence to the house as "b."

We know that a = 8 feet, and b = 3 feet. Our goal is to find the value of c.

By applying the Pythagorean theorem, we have the formula:

c^2 = a^2 + b^2

Substituting the values we know:

c^2 = 8^2 + 3^2
c^2 = 64 + 9
c^2 = 73

To find the length of the ladder (c), we can take the square root of both sides:

c = √(73)

Using a calculator, we find that √73 is approximately 8.54 feet.

Therefore, the length of the shortest ladder is approximately 8.54 feet.