I have another problem that I don't know how to solve. A car travels along a straight road, heading east for 1 hour, then traveling 30 minutes on another road that leads northeast. If the car had maintained a constant speed of 40 mph, how far is it from it's starting position? I kind of know how to set it up but I don't know how to find the angle to solve it, please explain! Thank you!
Starting from (0,0)
40mi/hr * 1 hr = 40 mi east, to (40,0)
40mi/hr * 1/2 hr = 20 mi NE adds another (20/√2,20/√2)=(14.14,14.14)
So, now the car is at (54.14,14.14)
The distance
d^2 = 54.14^2 + 14.14^2 = 3131.14
d = 55.96
You could also use the law of cosines. The angle between the two legs of the trip is 135°, so
d^2 = 40^2 + 20^2 - 2(40)(20)cos135°
= 1600+400-1131.37 = 3131.37
rounding from the approximations above accounts for the small difference.
1600+400+1131.37 = 3131.37
I'm confused with how you got 135 degrees? Please explain! Thank you for the help!
did you draw the diagram? Due east is along the x-axis. NE is at 45°.
So, the inside angle of the triangle is 180°-45° = 135°
If that kind of diagram gives you problems, you have a tough road ahead...
To solve this problem, we need to understand the concept of vector addition and use some trigonometry. Here's how you can approach it step-by-step:
Step 1: Start by drawing a diagram to visualize the problem. Draw a straight line to represent the initial one-hour journey along the east direction. Then draw a line at a 45-degree angle to represent the next 30-minute journey that leads northeast.
Step 2: Find the distance traveled for each leg of the journey. In the first hour, the car travels at a constant speed of 40 mph, so the distance covered is 40 miles. In the next 30 minutes, the car continues to travel at 40 mph, so the distance covered is (40 mph) * (0.5 hours) = 20 miles.
Step 3: Now, we need to find the net displacement of the car, not just the distance traveled. To do that, we'll use vector addition. Since the first leg of the journey is along a straight road, the direction of the displacement vector is east, and its magnitude is 40 miles.
Step 4: Now comes the trigonometry part. The next leg of the journey is at a 45-degree angle northeast. We can use the concept of right-angled triangles to find the eastward and northward components of this leg.
Step 5: Draw a right-angled triangle using the 45-degree angle line as the hypotenuse. The eastward component will be adjacent to the angle, and the northward component will be opposite to the angle.
Step 6: Since the hypotenuse of the triangle has a magnitude of 20 miles, using trigonometry, we can find the eastward and northward components. The cosine of 45 degrees is equal to the adjacent side divided by the hypotenuse. Therefore, the eastward component is (20 miles) * cos(45 degrees).
Step 7: Similarly, the sine of 45 degrees is equal to the opposite side divided by the hypotenuse. Therefore, the northward component is (20 miles) * sin(45 degrees).
Step 8: Now we have the eastward displacement of 20 miles and the net eastward displacement of 40 miles. To get the total eastward displacement, we need to add these two values together.
Step 9: Finally, you can use the Pythagorean theorem to calculate the distance of the car from its starting position. The Pythagorean theorem states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of its other two sides. In this case, the hypotenuse represents the total eastward displacement, and the other two sides represent the northward displacement (which is 0) and the eastward displacement (found in step 8).
By following these steps, you should be able to find the distance of the car from its starting position.