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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 $x=-4$.


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