Question

laura is sking on a circular ski trail that has a radius 1.4 km long. she starts at the 3 - oclock position (the easternmost part of the trail) and skis in the counterclockwise direction. imagine an angle with a vertex at the center of the circular trail that subtends the path laura has traveled.
a if laura has traveled 1.4 km along the ski trail then the angle has swept out 1.4 radian(s)
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• try again. start by drawing a diagram. how long is the subtended arc length? how long is the circles radius? how do their sizes compare and why is this comparison important?

b if laura has traveled 3 km along the ski trail then the angle has swept out (\\( \theta + 2 p \\)) + 3 radian(s)
preview

• try again. start by drawing a diagram. how long is the subtended arc length? how long is the circles radius? how do their sizes compare and why is this comparison important?

c complete the formula that expresses \\( \theta \\), the number of radians swept out by the angle, in terms of \\( d \\), the number of kilometers laura has traveled since she started sking.
\\( \theta = 1.4 \sin ( 2 p ) + 3 \\)
preview

• try again. start by drawing a diagram. how long is the subtended arc length? how long is the circles radius? how do their sizes compare and why is this comparison important?

Explanation:

Step1: Recall the arc - length formula

The arc - length formula is \(s = r\theta\), where \(s\) is the arc length, \(r\) is the radius of the circle, and \(\theta\) is the central angle in radians.

Step2: Solve for \(\theta\) when \(s = 1.4\) km and \(r = 1.4\) km

Substitute \(s = 1.4\) and \(r = 1.4\) into the formula \(s=r\theta\). We get \(\theta=\frac{s}{r}\).

$$ \theta=\frac{1.4}{1.4}=1 $$

Step3: Solve for \(\theta\) when \(s = 3\) km and \(r = 1.4\) km

Substitute \(s = 3\) and \(r = 1.4\) into the formula \(s = r\theta\). Then \(\theta=\frac{s}{r}\).

$$ \theta=\frac{3}{1.4}=\frac{30}{14}=\frac{15}{7}\approx2.14 $$

Answer:

a. \(1\) radian(s)
b. \(\frac{15}{7}\approx2.14\) radian(s)