To determine the length of the zipline wire, we can use the Pythagorean theorem. The situation can be visualized as a right triangle where:
- One leg (height of the building) is 20 meters,
- The other leg (horizontal distance to the island) is 50 meters,
- The hypotenuse (length of the zipline) is what we need to find.
According to the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
where \( c \) is the hypotenuse, and \( a \) and \( b \) are the two legs of the triangle.
Here, we have:
- \( a = 20 \) meters (height of the building),
- \( b = 50 \) meters (horizontal distance).
Substituting the values into the formula:
\[ c^2 = 20^2 + 50^2 \] \[ c^2 = 400 + 2500 \] \[ c^2 = 2900 \]
To find \( c \), we take the square root of 2900:
\[ c = \sqrt{2900} \]
Calculating that:
\[ c \approx 53.8516 \]
Rounding to the nearest tenth, the length of the zipline wire needs to be approximately 53.9 meters.