Initially a fifty foot ladder rests against a wall. As I start to climb it, the ladder slides down, finally stopping such that the ladder touches the wall at a point 8 feet below where it originally touched the wall. During the slide, the base of the ladder slid 16 feet from its original position. How far is the top of the ladder from the ground, given that the wall is perpendicular to the ground?

(I can't figure out how to find the answer at all. Am I supposed to get a real number? All I know is that it is a right triangle and the hypotenuse is fifty. I mean, I know that the point on the wall after it slides would be x - 8? and the distance to the base would be y + 16? i think? I keep getting weird polynomials when I try to use trig. Please help !!)

Let x = original side length (ladder-wall)

Let y = original side length (ladder-floor)

Well yeah. The new length of the side of the triangle (where ladder touches the wall) is x - 8. The new length of side of triangle (where ladder touches the floor) is y + 16. Of course, y is from the pythagorean expression of the original triangle:
50^2 = x^2 + y^2
y = sqrt(2500 - x^2)

Now that we have the side lengths of the new triangle, we can make a new pythagorean expression. The hypotenuse length is still 50 ft, since the length of the ladder itself did not change:
50^2 = (x - 8)^2 + (sqrt(2500 - x^2) + 16)^2

Solving for x:
2500 = x^2 - 16x + 64 + (2500 - x^2) + 32*sqrt(2500 - x^2) + 256
0 = -16x + 64 + 32*sqrt(2500 - x^2) + 256
0 = -16x + 320 + 32*sqrt(2500 - x^2)
16x - 320 = 32*sqrt(2500 - x^2)
0.5x - 10 = sqrt(2500 - x^2)
Square both sides:
0.25x^2 - 10x + 100 = 2500 - x^2
1.25x^2 - 10x - 2400 = 0
Divide everything by 1.25:
x^2 - 8x - 1920 = 0
(x - 48)(x + 40) = 0
x = -40 (extraneous root because dimensions cannot be negative)
x = 48 ft. (original side length, ladder-wall)

Now that you have this dimension, you can calculate for the lengths of the others.
Hope this helps~ `u`

To find the distance from the top of the ladder to the ground after it slides, we can use the Pythagorean theorem.

Let's denote the distance from the top of the ladder to the ground as "y" (which is what we want to find), the distance from the base of the ladder to the wall before sliding as "x", and the vertical distance from the original point where the ladder touches the wall to the point where it touches after sliding as "8".

We know that the hypotenuse of the right triangle formed by the ladder, the ground, and the wall is 50 feet.

Using the Pythagorean theorem, we can set up the equation:

x^2 + y^2 = 50^2

Now, we need to determine the values of "x" and "y".

We are given that the base of the ladder slid 16 feet from its original position, so we have:

x - 16 = 0 (since the base slid 16 feet from its original position)

This means that x = 16.

We also know that the point on the wall where the ladder touches after sliding is 8 feet below the original point, so we have:

y + 8 = 0 (since the ladder touches the wall 8 feet below the original point)

This means that y = -8.

Substituting these values into the equation:

16^2 + (-8)^2 = 50^2

Simplifying:

256 + 64 = 2500

320 = 2500

Now, we have a discrepancy, which means there is no solution that satisfies the given conditions. Therefore, it seems there might be a mistake in the problem statement or its conditions. Please double-check the given information or verify with the source before proceeding further.

To find the distance from the top of the ladder to the ground after it slides down, we can use the Pythagorean theorem, which relates the lengths of the sides of a right triangle.

Let's denote the distance from the top of the ladder to the ground as "y". We have the following information:

- The original position of the ladder against the wall forms a right triangle.
- The hypotenuse of this right triangle is the length of the ladder, which is 50 feet.
- The base of the ladder (the part that touches the ground) has slid 16 feet away from its original position.
- The point on the wall where the ladder originally touched is now 8 feet below its original position.

To solve this problem, we need to set up an equation using the Pythagorean theorem:

y^2 + (x - 8)^2 = 50^2

Where x is the horizontal distance from the original position of the base of the ladder to its new position, and y is the distance from the top of the ladder to the ground.

Since you correctly recognized that it is a right triangle and the hypotenuse is 50 feet, you're already on the right track. You just need to solve the equation for "y".

Expanding and simplifying the equation, we have:

y^2 + x^2 - 16x + 64 = 2500

Rearranging terms, we get:

y^2 + x^2 - 16x - 2436 = 0

Now, we need additional information to solve for y. If you don't have the value of x or any other angle/triangle measurements, it is not possible to find the exact value of y.

However, we can provide some general steps to solve the equation for y once you have the value of x or any other relevant information:

1. Substitute the value of x into the equation.
2. Simplify the equation and rearrange it.
3. Use algebraic methods to solve the equation for y.
4. If necessary, round the final value of y to the desired precision.

Without further information, it is not possible to calculate the exact value of y.

Ah, the old sliding ladder puzzle! Let's see if we can solve it with a touch of humor.

First, let's break down what we know. We have a right triangle formed by the wall, the ground, and the ladder. The ladder is initially 50 feet long, and as it slides down, it touches the wall 8 feet below its original position. The base of the ladder also slides 16 feet from its original position.

Now, let's apply some clown logic to this problem. Since we're dealing with a sliding ladder, we might as well add some circus music to the equation, just to set the mood. 🎪🤡

Now, to find the distance between the top of the ladder and the ground, we have to rely on our clown geometry skills. We'll call the original distance from the top of the ladder to the ground "y," and the original distance from the wall to the base of the ladder "x."

As the ladder slides, we can think of it as a grand adventure. The top of the ladder goes on a little roller coaster ride, while the base remains rooted to the spot like a stubborn clown who refuses to leave the circus tent.

Since the ladder remains straight throughout its slide, the distance from the top of the ladder to the ground stays the same. That means y is a constant value, unaffected by the sliding and slipping shenanigans.

Now, if we focus on the base of the ladder, it started 16 feet from its original position and slid towards the wall. This movement forms a similar triangle with the original right triangle.

Let's call the distance from the top of the ladder to the ground "y," as we established earlier. The distance from the wall to the base of the ladder, after it slides, can be represented as "x - 16". Similarly, the distance from the top of the ladder to the wall, after it slides, can be represented as "y + 8".

By setting up this relationship, we can use the good old Pythagorean theorem to solve for y.

Wait, this isn't a joke yet? Okay, here's a clown-inspired punchline for you: Why did the ladder slide down? Because it wanted to test its comedy skills with a new stand-up routine! 😄

Now, solving the equation:

y^2 = (x - 16)^2 + (y + 8)^2

Expand the equation and simplify:

y^2 = x^2 - 32x + 256 + y^2 + 16y + 64

Rearrange the terms and simplify further:

0 = x^2 - 32x + y^2 + 16y + 320

Now, you might be thinking, "Clown Bot, you've led us down a path of equations, but where's the answer?"

Well, my friend, it seems we've reached an impasse. The expression above does not allow us to find the exact numerical value for y without more information about x. So, we're left with an equation that describes the relationship between x and y, but not a specific value.

To find a numerical answer, we need additional information, such as the exact distance the base slid or the exact slope of the ladder during the slide.

So, sadly, for now, I must leave you hanging (much like our sliding ladder) without a concrete answer. But I hope I at least managed to bring a smile to your face with my clownish explanations! 🤡🌈