An angler hooks a trout and begins turning her circular reel at 1.5 rev/s. Assume the radius of the reel (and the fishing line on it) is 2 inches.
a. Let R equal the number of revolutions the angler has turned her reel and suppose L is the amount of line that she has reeled in. Find an equation for L as a function of R.

Verified step by step guidance

1. Understand that the problem involves a circular motion where the reel is turning, and we need to relate the number of revolutions (R) to the length of the line reeled in (L).
2. Recognize that each complete revolution of the reel corresponds to the circumference of the circle formed by the reel. The circumference C of a circle is given by the formula: C = 2πr, where r is the radius.
3. Substitute the given radius of the reel into the circumference formula. Here, the radius r is 2 inches, so the circumference C = 2π(2) = 4π inches.
4. Since each revolution reels in a length of line equal to the circumference of the reel, the amount of line L reeled in after R revolutions is L = R * C.
5. Substitute the expression for the circumference into the equation for L: L = R * 4π. Therefore, the equation for L as a function of R is L(R) = 4πR.

Key Concepts

Circumference of a Circle

The circumference of a circle is the distance around it, calculated using the formula C = 2πr, where r is the radius. In this context, the radius of the reel is 2 inches, so the circumference represents the length of fishing line that is reeled in with each complete revolution of the reel.

Linear Relationship

A linear relationship describes how one variable changes in relation to another. In this case, the amount of line L that is reeled in is directly proportional to the number of revolutions R made by the reel, which can be expressed as L = C * R, where C is the circumference of the reel.

Rate of Revolution

The rate of revolution indicates how quickly the reel is turned, measured in revolutions per second (rev/s). Here, the angler turns the reel at a rate of 1.5 rev/s, which helps determine how fast the line is being reeled in over time, linking the concepts of time, revolutions, and the length of line.