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Magazine Chapter 11: Problem 85
A fly is crawling along a wire helix so that its position vector is \(\mathbf{r}(t)=6 \cos \pi t \mathbf{i}+6 \sin \pi t \mathbf{j}+2 t \mathbf{k}, t \geq 0 .\) At what point will the fly hit the sphere \(x^{2}+y^{2}+z^{2}=100,\) and how far did it travel in getting there (assuming that it started when \(t=0\) )?
Short Answer
Expert verified
The fly hits the sphere at (6, 0, 8) and travels 24\(\pi\) units.
Step by step solution
01
Write Down the Condition for Intersection
The fly hits the sphere when its position satisfies the equation of the sphere. The equation of the sphere in our problem is \(x^2 + y^2 + z^2 = 100\). We need to plug in the components of \(\mathbf{r}(t)\) into this equation and find the values of \(t\) that satisfy it.
02
Substitute Components into Sphere Equation
The position vector is \(\mathbf{r}(t) = 6 \cos \pi t \mathbf{i} + 6 \sin \pi t \mathbf{j} + 2t \mathbf{k}\). Substituting these into the sphere equation gives:\[(6 \cos \pi t)^2 + (6 \sin \pi t)^2 + (2t)^2 = 100\]Simplifying the left hand side:\[36 (\cos^2 \pi t + \sin^2 \pi t) + 4t^2 = 100\]. Since \(\cos^2 \theta + \sin^2 \theta = 1\), the equation becomes:\[36 + 4t^2 = 100\].
03
Solve for t
Now, solve the equation \(36 + 4t^2 = 100\) for \(t\):\[4t^2 = 100 - 36\]\[4t^2 = 64\]\[t^2 = 16\]\[t = \pm 4\]Since \(t \geq 0\), we have \(t = 4\).
04
Calculate the Position at t=4
Plug \(t = 4\) into the position vector \(\mathbf{r}(t)\) to find the position at this time:\[\mathbf{r}(4) = 6 \cos 4\pi \mathbf{i} + 6 \sin 4\pi \mathbf{j} + 2(4) \mathbf{k}\]Evaluating the trigonometric functions:\[\cos 4\pi = 1, \quad \sin 4\pi = 0\]So, the position vector is:\[\mathbf{r}(4) = 6 \cdot 1 \mathbf{i} + 6 \cdot 0 \mathbf{j} + 8 \mathbf{k}\]\[= 6 \mathbf{i} + 8 \mathbf{k}\]
05
Calculate the Distance Traveled
To find the distance traveled by the fly, calculate the arc length of its path from \(t = 0\) to \(t = 4\). The arc length \(s\) of a vector function \(\mathbf{r}(t)\) is given by:\[s = \int_{0}^{4} \| \mathbf{r}'(t) \| \, dt \]The derivative \(\mathbf{r}'(t)\) is:\[\mathbf{r}'(t) = -6\pi \sin \pi t \mathbf{i} + 6\pi \cos \pi t \mathbf{j} + 2 \mathbf{k}\]Calculate the magnitude:\[\| \mathbf{r}'(t) \| = \sqrt{(-6\pi \sin \pi t)^2 + (6\pi \cos \pi t)^2 + 2^2}\]Simplification using \(\sin^2 \theta + \cos^2 \theta = 1\):\[= \sqrt{36\pi^2 + 4} = 6\pi \sqrt{1 + \frac{1}{9\pi^2}}\]Integrate to find the total arc length:\[s = \int_{0}^{4} 6\pi \, dt = 24\pi\]
06
Final Step: Conclusion
The fly hits the sphere at the coordinates \( (6, 0, 8) \), having traveled a distance of \(24\pi\) from its starting point.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Position Vector
A position vector is a mathematical tool used to specify the location of a point in space relative to an origin. In this problem, the fly's position on the helix is given by the vector function \[ \mathbf{r}(t) = 6 \cos \pi t \mathbf{i} + 6 \sin \pi t \mathbf{j} + 2t \mathbf{k} \].
- The coefficients of \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) indicate how far the fly is along the x, y, and z axes respectively.
- In our case:
- \(x = 6 \cos \pi t\): The x-component describes a circular motion.
- \(y = 6 \sin \pi t\): The y-component complements the x-component, completing the circular helix path.
- \(z = 2t\): This component shows linear movement upwards, adding a helical dimension to the path.
Sphere Equation
The sphere equation \[ x^2 + y^2 + z^2 = 100 \] represents a geometric shape in three-dimensional space.
- Here, \(x\), \(y\), and \(z\) are the Cartesian coordinates of any point on the sphere's surface.
- When a point (x, y, z) satisfies this equation, it lies exactly on the surface of the sphere.
Arc Length
Arc length is a concept from calculus used to measure the distance along a curve. Here, it's employed to find how far the fly travels on its helical path.
- The arc length formula for a curve defined by a vector function \(\mathbf{r}(t)\) is \[ s = \int_{a}^{b} \| \mathbf{r}'(t) \| \, dt \], where \(\| \mathbf{r}'(t) \|\) is the magnitude of the derivative of the position vector.
- The derivative \(\mathbf{r}'(t)\) represents the velocity vector, showing the fly's velocity at any point.
Integral Calculus
Integral calculus plays a pivotal role in these calculations by providing the tools to compute quantities like area and arc length.
- Integral calculus deals with the accumulation of quantities, which can relate to areas under curves or total distances traveled.
- In this problem, integration is used to compute the arc length of the helix, providing the exact path distance covered before hitting the sphere.
