8.5 A bunch of other stuff about radicals

Albert Einstein’s “Special Theory of Relativity” tells us that matter and energy are different forms of the same thing . (Previously, they were thought of as two completely different things.) If you have some matter, you can convert it to energy; if you have some energy, you can convert it to matter. This is expressed mathematically in the famous equation $E=m{c}^2$ , where $E$ is the amount of energy, $m$ is the amount of matter, and $c$ is the speed of light. So, suppose I did an experiment where I converted $m$ kilograms of matter, and wound up with $E$ Joules of energy. Give me the equation I could use that would help me figure out, from these two numbers, what the speed of light is.

The following figure is an Aerobie, or a washer, or whatever you want to call it—it’s the shaded area, a ring with inner thickness $r_1$ and outer thickness $r_2$ .

• A

  What is the area of this shaded region, in terms of $r_1$ and $r_2$ ?

• B

  Suppose I told you that the area of the shaded region is $32\mathrm{\pi }$ , and that the inner radius $r_1$ is 7. What is the outer radius $r_2$ ?

• C

  Suppose I told you that the area of the shaded region is $A$ , and that the outer radius is $r_2$ . Find a formula for the inner radius.

$\sqrt{100y}$

$\sqrt{\frac{x}{25}}$

$\sqrt{x+16}$

$\frac{\sqrt{50}}{2}=$

$5\sqrt{2}+2\sqrt{3}-3\sqrt{2}=$

$\sqrt{27}-\sqrt{48}=$

$\frac{1}{\sqrt{2}}+\frac{\sqrt{2}}{2}=$

• A

  Simplify. (Don’t use your calculator, it won’t help.)

• B

  Now, check your answer by plugging the original formula into your calculator. What do you get? Did it work?

$\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}=$

• A

  Simplify. (Don’t use your calculator, it won’t help.)

• B

  Now, check your answer by plugging the original formula into your calculator. What do you get? Did it work?

$\frac{1}{\sqrt{3}+1}-\frac{\sqrt{3}}{2}=$

• A

  Simplify. (Don’t use your calculator, it won’t help.)

• B

  Now, check your answer by plugging the original formula into your calculator. What do you get? Did it work?

Graph $y=\sqrt{x}$

• A

  Plot a whole mess of points. (Choose x-values that will give you pretty easy-to-graph y-values!)

• B

  What is the domain? What is the range?

• C

  Draw the graph.

Graph $y=\sqrt{x}-3$ by shifting the previous graph.

• A

  Plug in a couple of points to make sure your “shift” was correct. Fix it if it wasn’t.

• B

  What is the domain? What is the range?

Graph $y=\sqrt{x-3}$ by shifting the previous graph.

• A

  Plug in a couple of points to make sure your “shift” was correct. Fix it if it wasn’t.

• B

  What is the domain? What is the range?