Question

The outer edge of a playground slide is in the shape of a helix of radius 1.5 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vectorvalued function for the helix. Use a computer algebra system to graph your function. (There are many correct answers.)

Short answer

The vector-valued function of the helix that represents the outer edge of the slide is \( \mathbf{r}(t) = 1.5 \mathbf{i} \cos(t) + 1.5 \mathbf{j} \sin(t) + 2 \mathbf{k} t \). This can be graphed using a computer algebra system to visualize the playground slide.

Step-by-step solution

Formulating the equations of a helix

The general form of a vector-valued function for a helix is \( \mathbf{r}(t) = \mathbf{i} R \cos(t) + \mathbf{j} R \sin(t) + \mathbf{k} h t \) where \( R \) is the radius, \( h \) indicates the height and \( t \) is the parameter, typically associated with the angle subtended by the point on the helix from the positive x-axis. In this case, \( R = 1.5 \) meters, \( h = 2 \) meters/revolution.

Inserting Values into the Function

Plugging these values into the vector-valued function for a helix we get \( \mathbf{r}(t) = 1.5 \mathbf{i} \cos(t) + 1.5 \mathbf{j} \sin(t) + 2 \mathbf{k} t \). This is a vector-valued function that describes the outer edge of the slide.

Graphing the Function with a Computer Algebra System

Use graphing software or a computer algebra system to verify the shape of this function. As it is a helix, it should appear as a spiraling curve. This will allow for a visual representation of the playground slide.