To find the horizontal distance from the base of the tree to the point where the broken part touches the ground, we can model the situation as a right triangle. In this right triangle:
- The vertical side (the height from the base of the tree to the point of the break) is 4.1 ft.
- The hypotenuse (the length of the broken part of the tree) is 11.1 ft.
- We need to find the horizontal side, which we'll call \( x \) (the distance from the base of the tree to where the broken part touches the ground).
According to the Pythagorean theorem, which states that for a right triangle:
\[ a^2 + b^2 = c^2 \]
where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides, we can replace \( a \), \( b \), and \( c \) with our known values:
- Let \( a = 4.1 \) ft (vertical distance).
- Let \( c = 11.1 \) ft (hypotenuse).
- Let \( b = x \) (horizontal distance).
Thus, our equation becomes:
\[ (4.1)^2 + x^2 = (11.1)^2 \]
Calculating the squares:
\[ 16.81 + x^2 = 123.21 \]
Now, solve for \( x^2 \):
\[ x^2 = 123.21 - 16.81 \] \[ x^2 = 106.4 \]
Next, take the square root of both sides to find \( x \):
\[ x = \sqrt{106.4} \approx 10.32 \text{ ft} \]
Therefore, the horizontal distance from the base of the tree to the point where the broken part touches the ground is approximately 10.32 ft.