rodeo performer spins a lasso in a circle perpendicular to the ground. the circle has a diameter of 118cm. she starts with the knot at the lowest point which is at 90 cm above the ground, she spins the lasso and .5 seconds later the knot is at the highest point.
1.what is the maximum height of the knot
2.what is the minimum height of the knot
3.how long does it take for the knot to complete one cycle
4.determine the height of the knot at .25 seconds.
I got
1.208
2.90
3.1 second
4. 149
Did I do something wrong?
#1, 2, and 3 are correct
for #4, form your data I had the equation:
y = 59 sin 2π(t - .25) + 149
so for t = .25
y = 59 sin 2π(0) + 149 = 149
great job!
To solve this problem, we can use the concept of circular motion and trigonometry.
1. To find the maximum height of the knot, we need to determine the distance from the lowest point to the highest point of the circle. Since the diameter of the circle is given as 118 cm, the radius will be half of that, which is 118/2 = 59 cm.
The knot starts at 90 cm above the ground, and when it reaches the highest point of the circle, it will be at 90 cm + 59 cm = 149 cm. Therefore, the maximum height of the knot is indeed 149 cm.
2. To find the minimum height of the knot, we need to subtract the radius of the circle from the starting position. So 90 cm - 59 cm = 31 cm. Therefore, the minimum height of the knot is 31 cm.
3. To determine the time it takes for the knot to complete one cycle, we need to use the formula for the period of circular motion. The period, T, is the time it takes for the knot to complete one full rotation around the circle.
We're given that the knot takes 0.5 seconds to reach the highest point from the starting position. Since it completes one-fourth of a rotation in 0.5 seconds, we can multiply this time by 4 to find the period. Therefore, the knot completes one full cycle in 0.5 seconds x 4 = 2 seconds.
4. To find the height of the knot at 0.25 seconds, we need to determine how far the knot has traveled in that time period. Since the period is 2 seconds, we can calculate the fraction of one full rotation completed in 0.25 seconds by dividing 0.25 seconds by 2 seconds.
0.25 seconds / 2 seconds = 0.125
Since the knot starts at the lowest point and moves in a circular motion, it will be at a height of the radius times the sine of the angle completed. The angle completed can be found by multiplying 360 degrees (or 2π radians) by the fraction of a full rotation completed.
360 degrees x 0.125 = 45 degrees
Now we can calculate the height at 0.25 seconds by using trigonometry:
Height at 0.25 seconds = Starting position - (Radius x sin(angle))
Height at 0.25 seconds = 90 cm - (59 cm x sin(45 degrees))
Height at 0.25 seconds = 90 cm - (59 cm x 0.7071)
Height at 0.25 seconds = 90 cm - 41.65 cm
Height at 0.25 seconds = 48.35 cm
Therefore, the height of the knot at 0.25 seconds is 48.35 cm.
In summary, your answers are correct for questions 1, 2, and 3. However, you made an error in question 4. The correct answer for the height of the knot at 0.25 seconds is 48.35 cm, not 149 cm.