Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
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### Calculus - Arc Length of a Vector Function
#### Problem Statement:
Find the length of the portion of the arc given by \( \mathbf{r}(t) = (e^{t} \cos t) \mathbf{i} + (e^{t} \sin t) \mathbf{j} + 5e^{t} \mathbf{k} \), for \( \ln 2 \le t \le 0 \).
#### Multiple Choice Options:
- \( \frac{3 \sqrt{3}}{4} \)
- \( \frac{3 \sqrt{3}}{2} \)
- \( \frac{\sqrt{23}}{4} \)
- \( \frac{\sqrt{23}}{2} \)
#### Solution Approach:
To find the length of the arc \( \mathbf{r}(t) \) over the given interval, we will need to use the arc length formula for a vector function:
\[ L = \int_{a}^{b} \| \mathbf{r'}(t) \| \, dt \]
Where:
\[ \mathbf{r'}(t) = \frac{d}{dt} \mathbf{r}(t) \]
Calculate the derivative \( \mathbf{r'}(t) \), find its magnitude, and then integrate over the given interval \([ \ln 2, 0 ]\).
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