Question

Tom is standing 14 meters from the base of a kite. The kite string makes a 67degree angle with the ground. Find x, the length of the kite string. Round to the nearest hundredth

Answer 1

To find the length of the kite string \( x \) when Tom is standing 14 meters from the base of the kite, we can use trigonometric principles. Given that the kite string forms a 67-degree angle with the ground, we can use the cosine function.

The cosine function relates the adjacent side (the distance from Tom to the base of the kite) to the hypotenuse (the length of the kite string) as follows:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]

In this case:
- \(\theta = 67^\circ\)
- The adjacent side is \( 14 \, \text{m} \)
- The hypotenuse is \( x \)

Thus, we have:

\[
\cos(67^\circ) = \frac{14}{x}
\]

To find \( x \), we rearrange the equation:

\[
x = \frac{14}{\cos(67^\circ)}
\]

Now we compute \(\cos(67^\circ)\):

Using a calculator, we find:

\[
\cos(67^\circ) \approx 0.3907
\]

Now we substitute this value into the equation for \( x \):

\[
x \approx \frac{14}{0.3907} \approx 35.8
\]

Thus, rounding to the nearest hundredth gives us:

\[
x \approx 35.82 \, \text{meters}
\]

So the length of the kite string is approximately \( \boxed{35.82} \) meters.