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Chapter 4: Problem 26

Suppose an ant walks counterclockwise on the unit circle from the point (1,0) to the endpoint of the radius corresponding to \(130^{\circ} .\) How far has the ant walked?

Short Answer

Expert verified
The ant has walked approximately 2.85841 units along the circumference of the unit circle.

Step by step solution

01

Find the angle the ant has moved for to get to the endpoint

First of all, we'll need to understand that the ant starts at the point (-1,0) which corresponds to an angle of \(\pi\) radians (or 180掳). Knowing that the ant ends at 6 radians, we will subtract the initial angle to find the angle the ant has actually moved: angle moved = endpoint angle - initial angle angle moved = \( 6 - \pi \)
02

Find the proportion of the circle the ant has traversed

Now, we need to find out the proportion of the circle that the ant has traversed. We do this by dividing the angle moved by the total angle of a circle (which is \(2\pi \)): proportion = \(\frac{angle\:moved}{total\:angle\:of\:a\:circle} \) proportion = \(\frac{6 - \pi}{2\pi} \)
03

Use the circumference formula to find the total distance the ant has walked

Now that we have found the proportion of the circle that the ant has traversed, we can use the formula for the circumference of a circle to find the distance the ant has walked. The formula for the circumference of a circle is: C = \( 2\pi r \) Since it's a unit circle, the radius (r) is 1. Therefore, C = \(2\pi\). Now, let's find the proportion of the circumference the ant has walked by multiplying the circumference by the proportion: distance walked = proportion 脳 circumference distance walked = \((6 - \pi) \times \frac{1}{2\pi} \times 2\pi \)
04

Simplify and calculate the distance

As we can see, the \(2\pi\) will cancel out in the numerator and denominator: distance walked = \(6 - \pi \) Now, calculate the value: distance walked 鈮� \(6 - 3.14159 \) distance walked 鈮� 2.85841 Hence, the ant has walked approximately 2.85841 units along the circumference of the unit circle.

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