Natalie is skiing along a circular ski trail that has a radius of 2.6 km. she starts at the 3-o'clock position and travels in the ccw direction. natalie stops skiing when she is 0.942 km to the right and 2.423 km above the center of the ski trail. imagine an angle with its vertex at the center of the circular ski trail that subtends natalie's path.
a. how many radians has the angle swept out since natalie started skiing?
b. how many km has natalie skied since she started skiing?

Step-by-step explanation:

a. The tangent of the central angle is the ratio ...

  tan(θ) = (2.423 km)/(0.942 km) . . . . . definition of tangent

  θ = arctan(2.423/.942) ≈ 1.200 radians

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b. The length of the arc is ...

  s = rθ = (2.6 km)(1.2 radians) = 3.12 km

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The attached output from a graphics program shows the angle as 68.76°. Multiplying by π/180°, we can convert that to radians:

  θ = 68.76π/180 = 216.0/180 = 1.200 radians

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}

answer: kljdcdhjbdlchjad oac apoich a

step-by-step explanation: