a 15 foot ladder is placed against a vertical wall of a building with the bottom of the ladder stand on level ground 12 feet from the base of the building. How high up the wall does the ladder reach?

Well, I'm tempted to say that the ladder reaches the top of the wall, but I suppose that's not the answer you're looking for. Let me calculate it for you.

Using a good old trick from geometry called the Pythagorean theorem, we can figure out the height. So, the ladder, the wall, and the ground form a right-angled triangle. The length of the ladder (hypotenuse) is 15 feet, and the distance between the base of the building and the ladder is 12 feet (one side of the triangle).

Now, let's find the missing side, which represents the height up the wall. Using the Pythagorean theorem, we have:

a^2 + b^2 = c^2

Where a represents the height up the wall, b represents the distance between the base of the building and the ladder, and c represents the length of the ladder.

Plugging in the values:

a^2 + 12^2 = 15^2
a^2 + 144 = 225
a^2 = 225 - 144
a^2 = 81
a = √81
a = 9

So, the height up the wall that the ladder reaches is 9 feet.

Hope that helps, and stay safe!

To determine how high up the wall the ladder reaches, we can use the Pythagorean theorem, which states that for a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In this case, the ladder forms the hypotenuse of the right triangle, the distance from the base of the building to the ladder forms one side (12 feet), and the height up the wall forms the other side (let's call it h feet).

Using the Pythagorean theorem, we have:

hypotenuse² = base² + height²

15² = 12² + h²

225 = 144 + h²

Subtracting 144 from both sides:

81 = h²

Taking the square root of both sides:

√81 = √(h²)

9 = h

Therefore, the ladder reaches a height of 9 feet up the wall.

To find out how high up the wall the ladder reaches, we can use the Pythagorean theorem. The 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 is the hypotenuse, the height of the wall is one of the other sides, and the distance from the base of the building to the ladder is the remaining side. Let's label the height of the wall as 'h' and the distance from the base of the building to the ladder as 'd'.

According to the problem, we know the following:
- The length of the ladder (the hypotenuse) is 15 feet.
- The distance from the base of the building to the ladder is 12 feet.

We can now use the Pythagorean theorem:

ladder^2 = height^2 + distance^2

Substituting the given values:

15^2 = h^2 + 12^2
225 = h^2 + 144

Now, let's solve for 'h':

h^2 = 225 - 144
h^2 = 81

Taking the square root of both sides:

h = √81
h = 9

Therefore, the ladder reaches a height of 9 feet up the wall.

To recap, to determine the height, we used the Pythagorean theorem by squaring the length of the ladder and then subtracting the square of the distance from the base of the building to find the square of the height. Finally, we took the square root to find the height itself.

a^2 + b^2 = c^2

a^2 + 12^2 = 15^2

a^2 + 144 = 225

a^2 = 81

a = 9