Question

triangle:

15 degree
250 ft
x

A plane is taking off at a 15 degree angle from a horizontal runaway. When the altitude of the plane is 250 feet, apply trigonometric functions and the Pythagorean Theorem to find x, the horizontal distance the plane has flown. Round the answer to the nearest whole number.

x = __ feet

Answer 1

To find the horizontal distance \( x \) that the plane has flown when it is 250 feet in altitude at a 15 degree angle from the horizontal, we can use trigonometric functions.

In this scenario, we can use the sine function, which is defined for a right triangle as follows:

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

In our case:
- The angle \( \theta = 15^\circ \)
- The opposite side (altitude) is 250 feet
- The hypotenuse is the distance the plane has traveled

From the sine definition:
\[
\sin(15^\circ) = \frac{250}{\text{hypotenuse}}
\]

To find the hypotenuse (let's call it \( h \)):
\[
h = \frac{250}{\sin(15^\circ)}
\]

Now, we can calculate \( \sin(15^\circ) \):
\[
\sin(15^\circ) \approx 0.2588
\]

Now substituting this value into the equation for \( h \):
\[
h = \frac{250}{0.2588} \approx 966.2 \text{ feet}
\]

Now we can find the horizontal distance \( x \) using the cosine function:
\[
\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}
\]

Thus:
\[
\cos(15^\circ) = \frac{x}{h}
\]
where:
\[
h \approx 966.2 \text{ feet}
\]

Plugging this into the cosine formula,
\[
x = h \cdot \cos(15^\circ)
\]
Now we can calculate \( \cos(15^\circ) \):
\[
\cos(15^\circ) \approx 0.9659
\]

Substituting in to find \( x \):
\[
x = 966.2 \cdot 0.9659 \approx 933.5 \text{ feet}
\]

Finally, rounding to the nearest whole number, we have:
\[
x \approx 934 \text{ feet}
\]

Thus, the horizontal distance the plane has flown is \( \boxed{934} \) feet.