Question

Jack is flying his kite . He runs 100 feet away from his house and lets out 45 feet of string . The angle of elevation from the ground to the kite is 65 degrees . How far is the kite from the house ?

Answer 1

Answer: 125.80 ft

Step-by-step explanation:

Asuming the described situation is as shown in the figure below, we need to find the distance $h$ between the kite and Jacks house, but first we need to find the $x$, $y$ and then $d$.

How?

We will use trigonometry, especifically the trigonometric functions sine and cosine:

For $y$:

$sin(65 \°)=\frac{y}{45 ft}$ (1)

Where $y$ is the opposite side to the angle and $45 ft$ the hypotenuse.

Isolating $y$:

$y=40.78 ft$ (2)

For $x$:

$cos(65 \°)=\frac{x}{45 ft}$ (3)

Where $x$ is the adjacent side to the angle.

Isolating $x$:

$y=19.017 ft$ (4)

Finding $d$:

$d=x+100 ft$ (5)

$d=19.017 ft +100 ft$

$d=119.017 ft$ (6)

Now that we have found these values, we have to work with a bigger triangle, where the hypotenuse is the distance between the kite and Jack's house $h$ and the sides are the values calculated in (4) and (6).

So, in this case we will use the Pithagorean theorem:

$h^{2}=y^{2} +d^{2}$ (7)

Isolating $h$ and writing with the known values:

$h=\sqrt{y^{2} +d^{2}}$ (8)

$h=\sqrt{(19.017 ft)^{2} +(119.017 ft)^{2}}$ (9)

$h=125.80 ft$ This is the distance between the kite and the house