a 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?
h(t) = 149 - 59cos(2pi t)
all your answers are correct
1. To find the maximum height of the knot, we need to determine the radius of the circle formed by the spinning lasso. The diameter is given as 118 cm, so the radius would be half of that, which is 118/2 = 59 cm.
The maximum height of the knot is equal to the radius plus the initial height of the knot. Therefore, the maximum height would be 59 cm + 90 cm = 149 cm.
So, the maximum height of the knot is 149 cm.
2. The minimum height of the knot can be found by subtracting the radius from the initial height of the knot. Therefore, the minimum height would be 90 cm - 59 cm = 31 cm.
So, the minimum height of the knot is 31 cm.
3. To find the time it takes for the knot to complete one cycle, we need to calculate the period of the circular motion. The period is the time taken for one complete revolution, and it is given by the formula:
Period = 2 * pi * radius / velocity
Since the diameter is given as 118 cm, the radius would be 59 cm. We are not given the velocity, so we cannot determine the exact period.
4. To find the height of the knot at 0.25 seconds, we can assume that the knot moves uniformly in a circular motion. Therefore, we can use the equation for the height of a point on a circle as a function of time:
Height = (radius * sin(2 * pi * time / period)) + initial height
Since we don't know the period, we cannot calculate the height at 0.25 seconds accurately.
To solve this problem, you need to understand the basic principles of circular motion and the relationship between the radius of a circle and its height above the ground.
1. To find the maximum height, you can use the formula for the height of a circle above the ground at any point on its circumference. As the knot starts at the lowest point, which is 90 cm above the ground, and reaches the highest point when it completes half of the circle, you can calculate the radius of the circle by dividing the diameter by 2: 118 cm / 2 = 59 cm. The maximum height is then the sum of the radius and the initial height: 59 cm + 90 cm = 149 cm, not 208 cm. So, your answer is incorrect.
2. To find the minimum height, you simply subtract the radius from the initial height: 90 cm - 59 cm = 31 cm. Therefore, the minimum height is 31 cm.
3. To determine the time it takes for the knot to complete one cycle, you need to know the speed of the rope spinning. In this case, you are given that the knot takes 0.5 seconds to go from the lowest point to the highest point. Since this only covers half a cycle, you need to multiply it by 2 to get the full time: 0.5 seconds * 2 = 1 second. So, it takes 1 second for the knot to complete one cycle.
4. To find the height of the knot at 0.25 seconds, you can divide the total time for one cycle by 4, as 0.25 seconds represents a quarter of the cycle: 1 second / 4 = 0.25 seconds. Then, you can use the formula for the height at any given time during circular motion. Plug in the values: radius (59 cm) cos(2π * 0.25 / 1), where 2π is the angle in radians for one full cycle. Calculating this yields a height of approximately 133 cm, not 149 cm.
So, your answers for 1, 2, and 4 are incorrect. The correct answers are:
1. Maximum height of the knot: 149 cm
2. Minimum height of the knot: 31 cm
3. Time to complete one cycle: 1 second
4. Height of the knot at 0.25 seconds: 133 cm