Problem 85 - Calculus Early Transcendentals: Pearson New International Edition

Question

. 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 \(t=0\) )?

Step-by-Step Solution

Answer

The fly hits the sphere at (6, 0, 8) and travels about 72.32 units.

Step 1: Identify Intersection Condition

The fly will hit the sphere when its position vector satisfies the sphere's equation. We have the vector \( \mathbf{r}(t) = 6 \cos \pi t \mathbf{i} + 6 \sin \pi t \mathbf{j} + 2t \mathbf{k} \). The sphere's equation is \( x^2 + y^2 + z^2 = 100 \). To find the intersection, substitute \(x = 6 \cos \pi t \), \(y = 6 \sin \pi t \), and \(z = 2t \) into the sphere's equation and solve: \( (6 \cos \pi t)^2 + (6 \sin \pi t)^2 + (2t)^2 = 100 \).

Step 2: Simplify and Solve the Equation

Simplify the equation: \(36 \cos^2 \pi t + 36 \sin^2 \pi t + 4t^2 = 100\). Use the Pythagorean identity \( \cos^2 \pi t + \sin^2 \pi t = 1 \), which gives \(36 + 4t^2 = 100 \). Solve for \(t^2\): \(4t^2 = 64\), so \(t^2 = 16\), and thus \(t = 4\) (since \(t \geq 0\)).

Step 3: Find the Intersection Point

Once \(t = 4\), substitute back into the original position equation: \(x = 6 \cos (4\pi) = 6\), \(y = 6 \sin (4\pi) = 0\), \(z = 2 \times 4 = 8\). Therefore, the intersection point is \((6, 0, 8)\).

Step 4: Calculate the Distance Travelled

To compute the path length travelled by the fly, we need to calculate the arc length of the helix from \(t = 0\) to \(t = 4\). The arc length \( L \) of a curve \( \mathbf{r}(t) \) is given by \( L = \int_{0}^{4} \| \mathbf{r}'(t) \| \, dt\), where \( \mathbf{r}'(t) = \frac{d\mathbf{r}}{dt} \).

Step 5: Compute the Derivative of \(\mathbf{r}(t)\)

Find the derivative: \( \mathbf{r}'(t) = \frac{d}{dt}(6 \cos \pi t \mathbf{i} + 6 \sin \pi t \mathbf{j} + 2t \mathbf{k}) = -6 \pi \sin \pi t \mathbf{i} + 6 \pi \cos \pi t \mathbf{j} + 2 \mathbf{k}\).

Step 6: Compute the Magnitude of \(\mathbf{r}'(t)\)

The magnitude is \( \| \mathbf{r}'(t) \| = \sqrt{(-6 \pi \sin \pi t)^2 + (6 \pi \cos \pi t)^2 + (2)^2}\). Simplifying gives \( \| \mathbf{r}'(t) \| = \sqrt{36\pi^2 + 4} = \sqrt{36\pi^2 + 4} \).

Step 7: Integrate to Find the Arc Length

The arc length is \( L = \int_{0}^{4} \sqrt{36\pi^2 + 4} \, dt = \sqrt{36\pi^2 + 4} \times 4\) since the integrand is a constant. Calculate \( \sqrt{36\pi^2 + 4} \approx 18.08 \), so \( L \approx 18.08 \times 4 = 72.32\).

Key Concepts

Vector Calculus

Vector calculus is a field of mathematics that deals with vector fields and the differentiation and integration of vector functions. When we describe the movement of a fly along a wire helix, we are using a position vector. This vector, denoted as \( \mathbf{r}(t) \), allows us to capture the fly's position in space at any given time \( t \).

In the given exercise, the position vector is expressed as \( \mathbf{r}(t)=6 \cos \pi t \mathbf{i}+6 \sin \pi t \mathbf{j}+2t \mathbf{k} \). This representation describes a path in three-dimensional space, indicating how the fly moves along the helix as time progresses.

Working with vectors involves operations like differentiation. We compute the derivative of a vector function to find its velocity vector, \( \mathbf{r}'(t) \), which tells us how fast and in what direction the fly is moving along the helix. This vector plays a crucial role in calculating arc length, thereby helping us gauge the distance the fly travels over time.

Arc Length

The concept of arc length is crucial when measuring the distance along a curved path rather than a straight line. In this exercise, the fly's travel path forms a helical curve, so computing the arc length of this path allows us to determine the precise distance traveled.

To find the arc length of a curve represented by a vector function \( \mathbf{r}(t) \), we use the formula:

• \( L = \int_{a}^{b} \| \mathbf{r}'(t) \| \, dt \)

Here, \( \| \mathbf{r}'(t) \| \) is the magnitude of the derivative of the position vector, which represents the speed of the fly at time \( t \). In the case of the helix, finding \( \mathbf{r}'(t) \) and its magnitude is essential for computing this integral.

Once we determine the speed as a constant (specifically \( \sqrt{36\pi^2 + 4} \) in this case), the rest of the calculation simplifies to multiplying this constant by the interval length of travel time \([0, 4]\). This results in an arc length of approximately 72.32 units, demonstrating how far the fly travels along the helix to intersect the sphere.

Helix Equation

The helix, known for its spiral form, is a common structure in both natural formations and geometry. The equation provided for the fly's path is a classic representation of a helical structure in three dimensions. This path is expressed as a vector equation with trigonometric functions, effectively capturing the spiraling motion.

The helix equation \( \mathbf{r}(t)=6 \cos \pi t \mathbf{i}+6 \sin \pi t \mathbf{j}+2t \mathbf{k} \) shows:

• Horizontal circular motion with \( 6 \cos \pi t \) and \( 6 \sin \pi t \),
• Increasing elevation with the linear \( 2t \) component.

As \( t \) increases, the fly moves around the circle located in the \( xy \) plane while also ascending along the \( z \) axis, forming a three-dimensional spiral.

The helical motion is periodic due to the cosine and sine terms, which cause the fly to repeat its position cyclically in the \( xy \) plane every full rotation (corresponding to \( t \) values that are multiples of 2). Meanwhile, the linear function in the \( z \) direction enables the fly to gradually climb upwards, showcasing the dual nature of helical motion.

Geometry of Spheres

Spheres are fundamental geometric shapes defined by all points equidistant from a central point in three-dimensional space. In mathematical terms, a sphere's equation reads \( x^2 + y^2 + z^2 = r^2 \), where \( r \) is the radius of the sphere.

In this exercise, we are given a sphere with an equation \( x^2 + y^2 + z^2 = 100 \), implying a radius \( \sqrt{100} = 10 \). The fly's objective is to intersect this sphere while traveling along its helical path.

To find the intersection, we equate the fly's position vector components to the sphere's equation, resulting in a solvable equation for \( t \). This determines the exact point at which the helix pierces the sphere's surface, both spatially and temporally. Understanding this interaction emphasizes the spatial dynamics between linear paths and curved surfaces in geometry, highlighting the broader applicability of these mathematical constructs in real-world scenarios.