Instructions for finding a buried treasure includethe following: Go 736.4 paces at 130◦, turn to 199◦and walk 134 paces, then travel 157 paces at 195◦.
Find the magnitude of the resultant displacement from the starting point.
Answer in units of paces.
What is the direction of the resultant displacement?
Use counterclockwise from due East as
the positive angular direction, between the limits of −180◦ and + 180◦.
Answer in units of degrees.
Ahoy matey! Let's crunch those numbers and find that buried treasure!
To find the magnitude of the resultant displacement, we'll use a lil' trigonometry. First, we'll break down the distances and angles given:
Step 1: 736.4 paces at 130°
Step 2: Turn to 199°
Step 3: Walk 134 paces
Step 4: Travel 157 paces at 195°
Now, we'll calculate the horizontal and vertical components for each step:
Step 1: Horizontal = 736.4 * cos(130°)
Vertical = 736.4 * sin(130°)
Step 2: No movement, just a turn
Step 3: Horizontal = 134 * cos(0°)
Vertical = 134 * sin(0°)
Step 4: Horizontal = 157 * cos(195°)
Vertical = 157 * sin(195°)
Next, we'll sum up all the horizontal and vertical components to get the total displacement:
Total Horizontal Displacement = (736.4 * cos(130°)) + (157 * cos(195°))
Total Vertical Displacement = (736.4 * sin(130°)) + (157 * sin(195°))
Finally, we'll use the Pythagorean theorem to find the magnitude of the resultant displacement:
Magnitude = sqrt((Total Horizontal Displacement)^2 + (Total Vertical Displacement)^2)
Take a deep breath... and solve that equation!
Once you've got the magnitude of the resultant displacement, we can move on to finding the direction. We'll use a lil' bit of trig again:
Direction = arctan(Total Vertical Displacement / Total Horizontal Displacement)
Remember, we're using a counterclockwise angle from due East as the positive direction. And make sure to keep the answer within the limits of -180° to +180°.
Put on your treasure-hunting hat and crunch those numbers! Good luck, me matey!
To find the magnitude of the resultant displacement, we can use the Pythagorean theorem. The formula is:
Resultant displacement = √(x^2 + y^2)
1. Convert the paces and angles to x and y components.
- Starting with the first set of instructions: Go 736.4 paces at 130°
- x component = 736.4 * cos(130°)
- y component = 736.4 * sin(130°)
- Second set of instructions: turn to 199° and walk 134 paces
- x component = 134 * cos(199°)
- y component = 134 * sin(199°)
- Third set of instructions: travel 157 paces at 195°
- x component = 157 * cos(195°)
- y component = 157 * sin(195°)
2. Add up the x and y components obtained from each set of instructions.
- x_total = (x1 + x2 + x3)
- y_total = (y1 + y2 + y3)
3. Calculate the magnitude of the resultant displacement.
- Resultant displacement = √(x_total^2 + y_total^2)
4. Round the magnitude of the resultant displacement to the nearest whole number and provide the direction in degrees.
Note: Make sure to use a calculator that is set to degrees mode for the trigonometric calculations.
Let's calculate the values step-by-step:
First set of instructions: Go 736.4 paces at 130°
- x1 = 736.4 * cos(130°) ≈ -368.178 paces
- y1 = 736.4 * sin(130°) ≈ 661.261 paces
Second set of instructions: turn to 199° and walk 134 paces
- x2 = 134 * cos(199°) ≈ -36.468 paces
- y2 = 134 * sin(199°) ≈ -122.467 paces
Third set of instructions: travel 157 paces at 195°
- x3 = 157 * cos(195°) ≈ -146.940 paces
- y3 = 157 * sin(195°) ≈ 90.433 paces
Summing up the x and y components:
- x_total = (-368.178 + (-36.468) + (-146.940)) ≈ -551.586 paces
- y_total = (661.261 + (-122.467) + 90.433) ≈ 629.227 paces
Calculating the magnitude of the resultant displacement:
- Resultant displacement = √((-551.586)^2 + (629.227)^2) ≈ 830.337 paces
Rounded magnitude of the resultant displacement = 830 paces
To find the direction of the resultant displacement, we can use inverse trigonometric functions.
- Direction = atan2(y_total, x_total) ≈ 128.928°
Rounded direction of the resultant displacement = 129°
Therefore, the magnitude of the resultant displacement is 830 paces, and the direction is 129°.
To find the magnitude of the resultant displacement, we need to calculate the total distance covered in both the x-direction and y-direction.
Let's break down the given instructions step by step:
1. Go 736.4 paces at 130°:
- In the x-direction: distance covered = 736.4 * cos(130°)
- In the y-direction: distance covered = 736.4 * sin(130°)
2. Turn to 199° and walk 134 paces:
- In the x-direction: distance covered = 134 * cos(199°)
- In the y-direction: distance covered = 134 * sin(199°)
3. Travel 157 paces at 195°:
- In the x-direction: distance covered = 157 * cos(195°)
- In the y-direction: distance covered = 157 * sin(195°)
Now, let's add up the distances in the x-direction and y-direction:
Total x-direction distance = (736.4 * cos(130°)) + (134 * cos(199°)) + (157 * cos(195°))
Total y-direction distance = (736.4 * sin(130°)) + (134 * sin(199°)) + (157 * sin(195°))
To find the magnitude of the resultant displacement, we use the Pythagorean theorem:
Resultant displacement = sqrt((Total x-direction distance)^2 + (Total y-direction distance)^2)
This will give us the magnitude of the resultant displacement in units of paces.
To find the direction of the resultant displacement, we use trigonometry:
Direction = atan2(Total y-direction distance, Total x-direction distance) in the range of -180° to +180°.
This will give us the direction of the resultant displacement in degrees, measured counterclockwise from due East.
Perform the above calculations to find the magnitude and direction of the resultant displacement.
D = 736.4[130o] + 134[199o] + 157[195o].
X=736.4*Cos(130) + 134*Cos(199) + 157*Cos(195)
= -473.3 - 126.7 - 151.7 = -751.7 Paces.
Y=736.4*sin(130) + 134*sin(199) + 157*sin(195)
=564.1 - 43.63 - 40.63=479.8 Paces. Q2.
a. Magnitude=sqrt(X^2+Y^2) = 892 Paces.
b. Tan A = Y/X = 479.8/-751.7=-0.63833.
A = -32.6o = 32.6o N. of W. = 147.4o CCW
= The direction.