Question

Aubrey is flying a kite, holding her hands a distance of 2.5 feet above the ground and letting all the kite’s string play out. She measures the angle of elevation from her hand to the kite to be 25degrees
∘
. If the string from the kite to her hand is 150 feet long, how many feet is the kite above the ground? Round your answer to the nearest hundredth of a foot if necessary.

Answer 1

To determine how high the kite is above the ground, we can use trigonometry. The situation can be modeled as a right triangle where:

- The height of the kite above Aubrey's hands is the opposite side,
- The length of the string is the hypotenuse (150 feet),
- The angle of elevation from her hands to the kite is 25 degrees.

We can use the sine function, which is defined as:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

Here, \(\theta = 25^\circ\), the opposite side is the height of the kite above her hands (\(h\)), and the hypotenuse is 150 feet.

Rearranging the sine equation to find \(h\):

\[
h = \text{hypotenuse} \cdot \sin(\theta)
\]

Substituting the values:

\[
h = 150 \cdot \sin(25^\circ)
\]

Now, we can calculate \(\sin(25^\circ)\) using a calculator:

\[
\sin(25^\circ) \approx 0.4226
\]

Now, substituting this value back into the equation for \(h\):

\[
h \approx 150 \cdot 0.4226 \approx 63.39 \text{ feet}
\]

This \(h\) represents the height of the kite above her hands. Since her hands are 2.5 feet above the ground, we need to add this height to find the total height of the kite above the ground:

\[
\text{Total height above ground} = h + 2.5
\]
\[
\text{Total height above ground} \approx 63.39 + 2.5 \approx 65.89 \text{ feet}
\]

Therefore, the height of the kite above the ground is approximately **65.89 feet**.