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Magazine Chapter 17: Problem 78
Find \(c\) so that one revolution about the \(z\) axis of the helix gives an increase of \(\Delta z\) in the \(z\) -coordinate. $$x=2 \cos t, y=2 \sin t, z=c t, \Delta z=50$$
Short Answer
Expert verified
\[c = \frac{25}{\pi}\]
Step by step solution
01
Understand the Helix Equation
The parametric equations provided describe a helix: \(x = 2 \cos t\), \(y = 2 \sin t\), and \(z = ct\). This means that as the parameter \(t\) changes, the point revolves in a circle of radius 2 in the \(xy\)-plane while ascending along the \(z\)-axis at a rate determined by \(c\).
02
Identify the Condition for One Revolution
One revolution about the \(z\)-axis corresponds to a change of \(t\) from 0 to \(2\pi\). This interval represents a complete cycle through the values of \(x\) and \(y\) back to the starting point.
03
Calculate the Vertical Increase After One Revolution
During one revolution (\(t = 0\) to \(t = 2\pi\)), the change in the \(z\)-coordinate, \(\Delta z\), is given by the difference in \(z\) values at these \(t\) values: \[\Delta z = z(2\pi) - z(0) = c(2\pi) - c(0) = 2\pi c.\]
04
Set the Expression Equal to Given Increase and Solve for \(c\)
We are given that \(\Delta z = 50\). Set the expression from the previous step equal to 50:\[2\pi c = 50.\]Solve for \(c\):\[c = \frac{50}{2\pi} = \frac{25}{\pi}.\]
05
Conclusion
The value of \(c\) that makes one revolution about the \(z\)-axis result in a \(\Delta z\) of 50 is \(c = \frac{25}{\pi}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parametric Equations
Parametric equations are a powerful way to represent curves by defining the coordinates in terms of a single parameter, usually denoted as \( t \). They are widely used in multivariable calculus because they simplify the description of complex curves in three-dimensional space.
In this example, the helix is defined by three parametric equations: \(x = 2 \cos t\), \(y = 2 \sin t\), and \(z = ct\).
In this example, the helix is defined by three parametric equations: \(x = 2 \cos t\), \(y = 2 \sin t\), and \(z = ct\).
- Each equation describes how the coordinates \(x\), \(y\), and \(z\) change with the parameter \(t\).
- The equations for \(x\) and \(y\) create a circular path in the \(xy\)-plane with radius 2, while the \(z\)-equation indicates vertical motion.
- The parameter \(t\) acts as a time-like variable, moving the point along the path of the helix.
Helix
A helix is a type of three-dimensional curve that spirals around an axis while simultaneously moving along it. It resembles the shape of a spring or a screw. In our exercise, the helix is described using the parametric equations \(x = 2 \cos t\), \(y = 2 \sin t\), and \(z = ct\). What makes a shape a helix is:
- The revolution around a central axis, here, the \(z\)-axis.
- A continuous, consistent upward or downward motion along that axis.
- A constant circular motion in the plane perpendicular to the axis, which is the \(xy\)-plane in our case.
Z-axis Revolution
The concept of revolution in this context refers to the path a point takes around a fixed axis while staying at a constant distance from it. Specifically, it's about rotation around the \(z\)-axis.
- For a complete revolution, the parameter \( t \) varies from 0 to \( 2\pi \), which corresponds to a full circle in the \(xy\)-plane.
- This revolution alters the positions of \( x \) and \( y \) but returns to their original values, completing a circular loop.
Vertical Increase
Vertical increase refers to the change in the \( z \)-coordinate as the parameter \( t \) progresses. When dealing with a helix, this is how much the curve ascends (or descends) with each complete rotation around the axis.In the exercise, the vertical increase per revolution is denoted by \( \Delta z \). It is calculated using the difference in the \( z \)-values at the start and end of one revolution:
- If \( t \) goes from 0 to \( 2\pi \), then \( z \) changes from \( z(0) \) to \( z(2\pi) \).
- Our given condition is that \( \Delta z = 50 \).
- Plugging into the formula \( \Delta z = c (2\pi) \), we find that \( c \) must be set such that this equation holds true.
