Question

laura is skiing 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
◄ correct! the angle measure in radians tells us the subtended arc length measured in units of the radius length. so $\theta=\frac{1.4}{1.4}=1$. the subtended arc length is the same as the radius length, so the angle measures 1 radian.
◄ radian(s)
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b if laura has traveled 3 km along the ski trail then the angle has swept out
◄ correct! the angle measure in radians tells us the subtended arc length measured in units of the radius length. so $\theta=\frac{3}{1.4}\approx2.1429$. the subtended arc length is $\frac{3}{1.4}$ or about 2.1429 times as long as the circles radius, so the angle measures $\frac{3}{1.4}$ or about 2.1429 radians.
◄ radian(s)
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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 skiing.
◄ syntax error
◄ $\theta=$
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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?

Explanation:

Step1: Recall the arc - length formula

The formula for arc - length \(s = r\theta\), where \(s\) is the arc - length, \(r\) is the radius, and \(\theta\) is the central angle in radians. We can solve for \(\theta=\frac{s}{r}\).

Step2: Analyze part (a)

Given \(s = 1.4\) km and \(r = 1.4\) km. Substitute into \(\theta=\frac{s}{r}\), we get \(\theta=\frac{1.4}{1.4}=1\) radian.

Step3: Analyze part (b)

Given \(s = 3\) km and \(r = 1.4\) km. Substitute into \(\theta=\frac{s}{r}\), we get \(\theta=\frac{3}{1.4}=\frac{30}{14}\approx2.1429\) radians.

Step4: Analyze part (c)

From \(s = r\theta\), we can express \(\theta=\frac{s}{r}\), where \(s\) is the distance Laura has traveled (in kilometers) and \(r = 1.4\) km (the radius of the circular trail).

Answer:

a. \(1\) radian(s)
b. \(\frac{30}{14}\approx2.1429\) radian(s)
c. \(\theta=\frac{s}{1.4}\), where \(s\) is the distance Laura has traveled.