Wichita State University Logo

Math 344: Calculus III

Problem Set 1

1 Volume of a Solid¶

Let $B$ be a solid box with length $L$, width $W$ and height $H$. Let $S$ be the set of all points that are a distance of at most one from some point in $B$. Express the volume of $S$ in terms of $L$, $W$ and $H$.

2 True Direction¶

A sailboat is capable of sailing at a speed of $18\,\mathrm{km}/\mathrm{h}$ is still water. The captain heads due north according to the ship's compass in a lake with no measurable current. After 30 minutes the ship's gps indicates that, due to the wind, this ship has actually traveled $8\,\mathrm{km}$ in the direction $5^{\circ}\,\mathrm{E}$.

(a) What is the wind velocity?

(b) In what direction should the captain have traveled to reach the intended destination of $8\,\mathrm{km}$ due north.

3 Compute with Vectors¶

Suppose that $\mathbf{v}_1$ and $\mathbf{v}_2$ are vectors with $|\mathbf{v}_1|=2$, $|\mathbf{v}_2|=3$ and $\mathbf{v}_1\cdot\mathbf{v}_2=5$. Let $\mathbf{v}_3 = \text{proj}_{\mathbf{v}_1}\mathbf{v}_2$, $\mathbf{v}_4=\text{proj}_{\mathbf{v}_2}\mathbf{v}_3$, $\mathbf{v}_5=\text{proj}_{\mathbf{v}_3}\mathbf{v}_4$ and so on. Compute

$$ S = \displaystyle\sum_{n=1}^{\infty} \left|\mathbf{v}_n\right| $$

4 Describe (in words) and Sketch a Solid¶

A three dimensional solid sits in a three dimensional grid with its centroid located at the origin.

  • When illuminated by a light source on the $z$-axis, its shadow is a disk. By shadow we mean the shadow on a plane perpendicular to the indicated axis. For example if the light source is on the $z$-axis, and the three dimensional solid has height 2, then the light source is located at $(0,0,4)$ and the shadow appears on the plane $z=-4$.

  • When illuminated by a light source on the $y$-axis, its shadow is a square. By shadow we mean the shadow on a plane perpendicular to the indicated axis. For example if the light source is on the $y$-axis, and the three dimensional solid has width 2, then the light source is located at $(0,4,0)$ and the shadow appears on the plane $y=-4$.

  • When illuminated by a light source on the $x$-axis, its shadow is an isosceles triangle. By shadow we mean the shadow on a plane perpendicular to the indicated axis. For example if the light source is on the $x$-axis, and the three dimensional solid has length 2, then the light source is located at $(4,0,0)$ and the shadow appears on the plane $x=-4$.

Create a picture or diagram of the three dimensional solid that casts these three shadows on the planes $z=-4$, $y=-4$ and $x=-4$.

Your use of this self-initiated mediated course material is subject to our An international nonprofit organization that empowers people to grow and sustain the thriving commons of shared knowledge and culture. Creative Commons License 4.0