To find the length of the kite string \( x \), we can use trigonometry. Specifically, since we know the distance from Tom to the base of the kite (14 meters) and the angle (67°), we can use the tangent function, which is defined as the opposite side over the adjacent side in a right triangle.
In this scenario:
- The opposite side is the height of the kite, which we don't know.
- The adjacent side is the distance from Tom to the base of the kite, which is 14 meters.
- The angle is 67°.
Using the tangent function:
\[ \tan(67°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{14} \]
Where \( h \) is the height of the kite.
To find \( h \):
\[ h = 14 \cdot \tan(67°) \]
Next, we need to find the length of the kite string \( x \). In a right triangle, \( x \) can be found using the Pythagorean theorem:
\[ x = \sqrt{(14^2 + h^2)} \]
First, we calculate \( h \):
- Calculate \( \tan(67°) \):
\[ \tan(67°) \approx 2.35585 \]
- Calculate \( h \):
\[ h = 14 \cdot 2.35585 \approx 33.0 \text{ meters} \]
Now, we find \( x \) (the kite string length):
- Calculate \( x \):
\[ x = \sqrt{(14^2 + h^2)} = \sqrt{(14^2 + (33.0)^2)} = \sqrt{(196 + 1089)} = \sqrt{1285} \approx 35.83 \text{ meters} \]
Thus, the length of the kite string rounded to the nearest hundredth is:
\[ \boxed{35.83 , \text{m}} \]