a ladder reaches 12m up to a vertical wall and has a gradient of 4. how far is the bottom of the ladder from the wall

To find the distance between the bottom of the ladder and the wall, we can use trigonometry.

The gradient of the ladder, which is the ratio of the vertical change to the horizontal change, is given as 4.

Let's assume the vertical change is 4x and the horizontal change is x. Since the ladder reaches 12m up to the wall, we can write the equation:

4x = 12

Dividing both sides of the equation by 4, we find:

x = 3

Therefore, the horizontal distance, which is the distance between the bottom of the ladder and the wall, is 3 meters.

To find the distance between the bottom of the ladder and the wall, we can use the Pythagorean Theorem, which states that in a right 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 forms the hypotenuse, the vertical wall is one of the sides, and the distance between the bottom of the ladder and the wall is the other side.

Let's denote the distance between the bottom of the ladder and the wall as "x". According to the Pythagorean Theorem, we have:

x^2 + 4^2 = 12^2

x^2 + 16 = 144

x^2 = 144 - 16

x^2 = 128

To find the value of x, we take the square root of both sides:

x = √128

Simplifying the square root of 128, we get:

x ≈ 11.314

Therefore, the distance between the bottom of the ladder and the wall is approximately 11.314 meters.

oops - I misread the question -- I was thinking the ladder was 12m long.

Since the gradient is 4, the ladder goes 4 times as far up the wall as it reaches away from the wall at the base.

Even easier...

well, x^2 + (4x)^2 = 12^2