Vanessa is skiing along a circular ski trail that has a radius of 2.5 km. She starts at the 3-o'clock position and travels in the CCW direction. Vanessa stops skiing when she is 1.244 km to the right and 2.169 km above the center of the ski trail. Imagine an angle with its vertex at the center of the circular ski trail that subtends Vanessa's path.How many radians has the angle swept out since Vanessa started skiing?

Answers

Answer 1

Answer:

Answer:

1.05 radians

Step-by-step explanation:

Given that Vanessa is skiing along a circular ski trail that has a radius of 2.5 km. She starts at the 3-o'clock position and travels in the Counter Clockwise direction.

She stops skiing when she is 1.244 km to the right and 2.169 km above the center of the ski trail.

This can be represented as (1.244, 2.169) on a circle of radius 2.5 km.

From the coordinate point (1.244, 2.169) derived, x=1.244 and y=2.169.

By the definition of tangent,

$\tan \theta =(y)/(x) \n\n\tan \theta =(2.169)/(1.244)\n\n \theta=\arctan (2.169)/(1.244)\n\n\n \theta=1.05006$

Vanessa swept out approximately 1.05 radians since she started skiing.

Answer 2

Answer:

Final answer:

Vanessa's path on the circular ski trail has swept out an angle of 1.076 radians since she started skiing.

Explanation:

First, let's find the position vector of Vanessa's stop point. The position vector is given by the coordinates (x, y), where x is the horizontal distance and y is the vertical distance from the center of the ski trail. In this case, x = 1.244 km to the right and y = 2.169 km above the center. Therefore, the position vector is (1.244 km, 2.169 km).

Next, we need to find the angle swept out by Vanessa's path. To do this, we can use trigonometry. The radius of the ski trail is 2.5 km, and the position vector is (1.244 km, 2.169 km). We can use the inverse tangent function to find the angle:

angle = tan^(-1)(y/x) = tan^(-1)(2.169/1.244) = 61.63 degrees

Finally, we need to convert the angle from degrees to radians. To convert degrees to radians, we use the conversion factor π/180. So, the angle in radians is:

angle_radians = 61.63 degrees * (π/180) = 1.076 radians

Learn more about Swept Angle here:

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