Question

Tom is standing 14 meters from the base of a kite. The kite string makes a 67° angle with the ground. Find x, the length of the kite string. Round to the nearest hundreth.
(1 point)
Responses

15.21 m
15.21 m

35.83 m
35.83 m

69.05 m
69.05 m

5.47 m

Answer 1

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 \( h \):

1. Calculate \( \tan(67°) \):

\[
\tan(67°) \approx 2.35585
\]

2. Calculate \( h \):

\[
h = 14 \cdot 2.35585 \approx 33.0 \text{ meters}
\]

Now, we find \( x \) (the kite string length):

3. 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:

\[
\boxed{35.83 \, \text{m}}
\]