A 19

​-foot
ladder is placed against a vertical wall of a​ building, with the bottom of the ladder standing on level ground 13

feet from the base of the building. How high up the wall does the ladder​ reach?

Well, it seems like the ladder is playing a classic game of "How High Can I Climb?" Though I must say, it's a little reluctant to reveal its exact reach.

But fear not! We can solve this riddle using a bit of math. The ladder, being 19 feet long, forms a right triangle with the ground and the wall. The distance from the base of the building to where the ladder touches the wall is 13 feet.

Now, let's channel our inner Pythagoras and do some calculation magic. According to the Pythagorean Theorem (a^2 + b^2 = c^2), where the ladder is the hypotenuse (c), and the base of the building is one side (a), and the height on the wall is the other side (b), we can solve for b.

So, b^2 = c^2 - a^2
b^2 = 19^2 - 13^2
b^2 = 361 - 169
b^2 = 192

Therefore, the ladder reaches the lofty height of the square root of 192 feet up the wall. But since I don't want to spoil all the fun, let's simply say that it reaches pretty high up!

To find the height on the wall that the ladder reaches, we can use the Pythagorean theorem. The theorem states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse. In this case, the ladder is the hypotenuse, and the distance from the base of the building to the wall is one leg.

Given:
Length of the ladder (hypotenuse) = 19 feet
Distance from the base of the building to the wall (leg) = 13 feet

Using the Pythagorean theorem, we can solve for the height on the wall that the ladder reaches.

Let's denote the height on the wall that the ladder reaches as "h".

From the theorem, we have:
(h^2) + (13^2) = (19^2)

Simplifying the equation:

h^2 + 169 = 361

Subtracting 169 from both sides:

h^2 = 361 - 169
h^2 = 192

Taking the square root of both sides:

√(h^2) = √192
h = √192

Calculating the square root of 192:

h ≈ 13.86

Therefore, the ladder reaches approximately 13.86 feet up the wall.

To find out how high up the wall the ladder reaches, we can use the Pythagorean theorem. The Pythagorean theorem states that in a 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, the ladder is the hypotenuse of the right triangle, and its length is 19 feet. One side of the right triangle is the vertical distance from the base of the ladder to where it reaches the wall, which we will call "x". The other side of the right triangle is the horizontal distance from the base of the ladder to the building, which is 13 feet.

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

19^2 = 13^2 + x^2

Simplifying the equation:

361 = 169 + x^2

Subtracting 169 from both sides of the equation:

192 = x^2

Taking the square root of both sides of the equation:

x = √192

Using a calculator, we find that the square root of 192 is approximately 13.86.

Therefore, the ladder reaches approximately 13.86 feet up the wall.

a^2 + b^2 = c^2

13^2 + b^2 = 19^2

169 + b^2 = 361

b^2 = 192

b = 13.856 feet