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Math 243: Calculus II

Problem Set 1

Problem Solving¶

The following problems are meant to test and challenge your problem-solving skills. Looking up, buying, or copying the answers won't help you pass your calculus class. Everyone can use the online Discussion Forum to discuss the answers, however do not publish the solutions. Some of them might require considerable time to think through. We can discuss online,

The Four Principles of Problem Solving¶

  • Understand the Problem
  • Think of a Plan
  • Carry out the Plan
  • Look Back

We an do this without outlining the solution online. If you get stuck, join the online discussion. You can use Desmos, Geogebra, or your favorite software to create graphs. List your answers in numerical order.

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 one unit from the closest 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 Show that the planes¶

$$ x + y - z = 1\text{ and }2x - 3y + 4z = 5 $$

are neither parallel, nor perpendicular using dot product and cross product.

(b) Compute the angle between these two planes.

(c) Compute the distance between the point $(3,0,2)$ and the plane $2x - 3y + 4z = 5$.

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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