Henry walks out of his hall and proceeds to walk 160 feet due north. He then turns left (west) and walks 150 more feet. Finally, he turns 65 degrees left and walks 280 feet. How many feet is Henry from his starting point?
his final location, relative to the starting point is
(0,160) + (-150,0) + (-280cos65°,-280sin65°)
Add up all the pairs and then just use the distance formula.
Well, Henry seems to be quite the wanderer! Let's see how far he ended up. To visualize it, he walked 160 feet north, then 150 feet west, and finally, 280 feet at an angle of 65 degrees. Now, if we were to draw this path on a map, it would look like a zigzag drawn by a drunken spider.
But fear not! With a bit of math magic, we can calculate the total distance Henry traveled. Using the Pythagorean theorem, we can find that the distance Henry traveled north and west forms a right triangle. So, using the formula a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse, we can calculate the distance.
The distance Henry traveled north and west can be calculated as √(160² + 150²), which results in approximately 212.68 feet.
Next, we need to find out how much distance he traveled at an angle. If we consider this distance as the third side of a triangle, we can calculate the total distance he traveled using the same formula: a² + b² = c². This time, a and b are the distances he traveled north and west (212.68 feet) and c will be the total distance he traveled.
Plugging in the values, we have √(212.68² + 280²), which results in approximately 355.98 feet.
So, to answer your question, Henry is approximately 355.98 feet away from his starting point. I hope Henry remembered to bring his GPS!
To find out how many feet Henry is from his starting point, we can use the concept of vector addition. Let's break down Henry's movements step-by-step:
1. First, Henry walks 160 feet due north.
2. Then, he turns left (west) and walks 150 more feet.
3. Finally, he turns 65 degrees left and walks 280 feet.
To calculate the final displacement, we'll need to find the north and west components separately and then add them up.
Step 1: North component
Henry walks 160 feet due north. This means his north component is 160 feet.
Step 2: West component
Henry turns left (west) and walks 150 feet. This means his west component is 150 feet.
Step 3: Displacement due to turning left
Henry turns 65 degrees left and walks 280 feet. To calculate the displacement from this movement, we can use trigonometry:
Vertical component: 280 feet * sin(65 degrees) ≈ 246.67 feet
Horizontal component: 280 feet * cos(65 degrees) ≈ 124.03 feet
Now, we can add up the north, west, and turning displacement components:
North component: 160 feet
West component: 150 feet
Vertical component: 246.67 feet
Horizontal component: 124.03 feet
Adding the north and vertical components: 160 feet + 246.67 feet ≈ 406.67 feet
Adding the west and horizontal components: 150 feet + 124.03 feet ≈ 274.03 feet
Now, we have the total north and west components. To find the total displacement, we can use the Pythagorean theorem:
Total displacement = √((North component)^2 + (West component)^2)
Total displacement = √((406.67 feet)^2 + (274.03 feet)^2)
Total displacement ≈ √(165335.59 feet^2 + 75169.20 feet^2)
Total displacement ≈ √(240504.79 feet^2)
Total displacement ≈ 490.36 feet
Therefore, Henry is approximately 490.36 feet from his starting point.
To find the distance between Henry's starting point and his final position, we can use the concept of vector addition. We will break down his movements into separate vectors and add them up.
First, let's label Henry's starting position as point A.
Next, Henry walks 160 feet due north. Walking due north means moving directly up, along the y-axis. Therefore, this vector can be represented as Ay, where Ay = 160 feet.
Henry then turns left (west) and walks 150 feet. Walking west means moving directly to the left, along the x-axis. This vector can be represented as Ax, where Ax = -150 feet because Henry is moving in the negative x-direction.
Finally, Henry turns 65 degrees left and walks 280 feet. To determine the direction of this vector, we need to take into account Henry's new heading after the previous movements. Since he started by walking north (positive y-direction) and then turned left (west), he is currently facing the south (negative y-direction). After turning 65 degrees left from the south, he will be facing the east (positive x-direction). So, this vector can be represented as Bx, where Bx = 280 * cos(65°) = 113.491 feet, and By, where By = 280 * sin(65°) = 251.545 feet.
Now, to determine the total displacement, we add the x- and y-components of all vectors:
Total displacement in x-direction = Ax + Bx = -150 feet + 113.491 feet = -36.509 feet (rounded to three decimal places)
Total displacement in y-direction = Ay + By = 160 feet + 251.545 feet = 411.545 feet (rounded to three decimal places)
Using the Pythagorean theorem, we can find the magnitude of the displacement:
Magnitude = √((-36.509 feet)^2 + (411.545 feet)^2)
= √(1332.749 + 169410.704)
= √170743.453
≈ 413.261 feet
Therefore, Henry is approximately 413.261 feet away from his starting point.