Question

The upper part of a tree is broken by wind and falls to the ground without being detached. If the top of the broken part touches the ground at a distance of 12 feet from the foot of the tree, calculate the height at which it is broken if the original height is 24 feet.

Answer 1

Let h be the height at which the tree is broken, in feet.

Let l be the length of the broken part of the tree, in feet.

The original height of the tree is 24 feet.

The length of the broken part, l, is equal to the original height minus the height at which it is broken:

l = 24 – h  ....(1)

A right-angled triangle is formed by the standing part of the tree, the distance from the foot of the tree to where the broken part touches the ground and the broken part itself. 

The sides of this right-angled triangle are h (height of the standing part), 12 feet (distance from the foot) and l (hypotenuse, the broken part).

The Pythagorean theorem is applied:

h² + 12² = l²

Putting the value of l in the equation (1),

h² + 12² = (24 – h)²

Expand the equation:

h² + 144 = 576 – 48h + h²

Subtract h² from both sides:

144 = 576 – 48h

Rearrange the equation to solve for h:

48h = 576 – 144

Calculate the value:

48h = 432

Divide to find h:

h = `432/48`

h = 9

∴ The height at which the tree is broken is 9 feet.