To determine the horizontal distance from Julia to the boat, we use trigonometric relationships in a right triangle. Let's outline the steps to solve this problem:
1. Understanding the Triangle:
Julia is parasailing, and the cable attaching her to the boat makes a right triangle with the ocean surface. In this triangle:
- The hypotenuse is the cable, which measures 500 feet.
- The angle of depression (from Julia's line of sight to the boat) is given as 41°.
2. Relating the Angle of Depression:
The angle of depression from Julia to the boat is equivalent to the angle of elevation from the boat to Julia. This is because the angles of elevation and depression are congruent when observed from corresponding alternate positions.
3. Trigonometric Function:
To find the horizontal distance (adjacent side of the right triangle), we use the cosine function, which relates the adjacent side and the hypotenuse in a right triangle:
[tex]\[
\cos(\text{angle}) = \frac{\text{adjacent}}{\text{hypotenuse}}
\][/tex]
4. Setting Up the Equation:
Let [tex]\( d \)[/tex] be the horizontal distance.
We know:
[tex]\[
\cos(41°) = \frac{d}{500}
\][/tex]
5. Solving for [tex]\( d \)[/tex]:
Multiply both sides by 500 to isolate [tex]\( d \)[/tex]:
[tex]\[
d = 500 \times \cos(41°)
\][/tex]
6. Calculating the Distance:
By using a calculator or looking up the value of [tex]\(\cos(41°)\)[/tex], which is approximately 0.7547:
[tex]\[
d = 500 \times 0.7547
\][/tex]
7. Result:
[tex]\[
d = 377.354790111386
\][/tex]
Therefore, the horizontal distance from Julia to the boat is approximately 377.35 feet.
