One day you go hiking at a nearby nature preserve. At first, you follow the stra

Question

One day you go hiking at a nearby nature preserve. At first, you follow the straight, clearly marked trails. From your starting point, you travel 2.00 miles down the 1st trail. Then you turn to your left by 30.0° to follow a 2nd trail for 1.60 miles. Next, you turn to your right by 160° and follow a 3rd trail for 2.20 miles. At this point you are getting very tired and would like to get back as quickly as possible, but all of the available trails seem to lead you deeper into the woods. You would like to take a shortcut directly through the woods (ignoring the trails). How far to your right should you turn, and how far do you have to walk, to go directly back to your starting point? Feel free to use the provided vector drawing board to help visualize your work.

Map One day you go hiking at a nearby nature preserve. At first, you follow the straight, clearly marked trails. From our starting point, you travel 2.00 miles down the 1st rail. Then you turn to your left by 30.0 to follow a 2nd trail for 1.60 miles. Next, you turn to your right by 160° and follow a 3rd trail for 2.20 miles. At this point you are getting very tired and would like to get back as quickly as possible, but all of the available trails seem to lead you deeper into the woods. You would like to take a shortcut directly through the woods (ignoring the trails). How far to your right should you turn, and how far do you have to walk, to go directly back to your starting point? d1 Feel free to use the provided vector drawing board to help visualize your work 0.25 miles Number to the right, Note: The vector drawing board is meant as a o Turn uide and a reality check, but may not give you the answers with sufficient precision. It is preferable to calculate the magnitude and direction by hand Number then walk miles

Explanation / Answer

Let us assume the first path points in the direction of the +x-axis.

So, at the end of first path:
Ax = 2.00 mi

and at the end of second path:
Ax = 2.00 + 1.60cos30.0° mi
Ax = 3.386 mi

Ay = 1.60 * sin30.0° mi
Ay = 0.80 mi

Now at the end of third path:
Ax = 3.386 + 2.20cos(30.0° - 160°) mi
Ax = 1.972 mi

Ay = 0.80 + 2.20sin(30.0° - 160°) mi
Ay = -0.885 mi

Now the reference angle to help you get back to where you started:
tan = Ay / Ax
tan = (-0.885) / (1.972)
= -24.1° (24.1° clockwise from +x-axis)

Means the angle from + x -axis in anti-clockwise direction = 180 - 24.1° = +155.9°

And you're facing 30.0° - 160° = -130° (or 360° - 130° = +230°) at the end of the third path,

Therefore, you'll need to turn
230° - 155.9° = 74.1° to the right (Answer of first part).

And walk a distance:
A = sqrt[(Ax)² + (Ay)²]
A = sqrt[(1.972 mi)² + (-0.885 mi)²]
A = 2.16 mi

This is the answer of second part.

Related Questions

Navigate

• Browse (All)
• Browse O
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.