Calculate the arc length of the indicated portion of the curve r(t). r(t) = (4t sin t + 4 cos t)i + (4t cos t – 4 sin t)j; - 2sts6 O A. 160 О В. 80 ОС. 64 O D. 128
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.
Need help with this arc length curve calc problem
![### Calculating the Arc Length of a Curve
#### Problem Statement:
Calculate the arc length of the indicated portion of the curve \(\mathbf{r}(t)\).
\[
\mathbf{r}(t) = (4t \sin t + 4 \cos t) \mathbf{i} + (4t \cos t - 4 \sin t) \mathbf{j}, \quad -2 \leq t \leq 6
\]
#### Options:
- **A. 160**
- **B. 80**
- **C. 64**
- **D. 128**
This problem requires understanding of calculating arc length using parameterized curves. The given vector function \(\mathbf{r}(t)\) describes a path in the plane. The arc length \(L\) of the curve from \( t = a \) to \( t = b \) can be found using the integral:
\[
L = \int_{a}^{b} \|\mathbf{r}'(t)\| \, dt
\]
where \(\|\mathbf{r}'(t)\|\) is the magnitude of the derivative of \(\mathbf{r}(t)\).
Do note that proper steps and consideration of derivatives and integration are involved in solving this problem.
### Important Considerations:
- Calculate \(\mathbf{r}'(t)\) with respect to \(t\).
- Find the magnitude \(\|\mathbf{r}'(t)\|\).
- Integrate \(\|\mathbf{r}'(t)\|\) within the given limits to find \(L\).
Use the above strategies to solve the problem and match your final answer with one of the provided options.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb1b08b0-0f3e-453c-902f-85fb7d668dbb%2Fdfb38447-4b30-4601-a380-71a8bd51ca9c%2F4fpck0pj_processed.png&w=3840&q=75)
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