Radical equations
Applying Radical Equations 2 Applying Radical Equations 2
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- A bead artist drills a tiny hole from the top corner of a
- cube bead to the opposite bottom corner of that bead.
- It's a cube bead.
- Each edge of the bead is 1/2 centimeters.
- How long is the hole that she drilled?
- So if I'm interpreting this properly, let me draw that
- cube bead, which I will just draw as a
- cube, a big cube here.
- Nicer cubes have been drawn in the history of cube drawing,
- but I'll try my best. So let's see.
- This is my best shot at a freehand cube.
- I think that does the job reasonably well.
- If this cube was transparent, you could even imagine kind of
- that backside over there, and the base of the cube if this
- was a transparent cube.
- They tell us that each edge of this cube bead-- they call
- this a cube bead-- each edge of the cube bead is 1/2
- centimeter.
- So this is 1/2 centimeters, this is 1/2 centimeters, and
- that is 1/2 centimeters.
- And then they say that they drilled a hole from the top
- corner of a cube bead to the opposite bottom
- corner of the bead.
- From the top corner to the opposite bottom.
- So I think they're talking about the longest diagonal
- that can fit in the cube.
- Going from the back, right top corner, over here, to the
- front, left bottom corner, right over there, that's the
- longest diagonal that could fit.
- So let me draw that.
- So it would look something like this.
- This is the distance in question.
- How long is the hole that she drilled?
- That is the distance that we have to figure out.
- And let's see how we can think about that.
- And this is a bit of a classic in terms of
- figuring out a distance.
- The longest diagonal of a cube or some type of
- a rectangular prism.
- And the trick here is to see that this is the hypotenuse of
- a triangle, although the triangle isn't completely
- obvious to you just yet.
- And to think about this as the hypotenuse of a right
- triangle, what you have to do is visualize the diagonal of
- the bottom face of this cube.
- Think about this diagonal right here that I'm drawing.
- So the diagonal of the bottom face of this cube.
- If you think of it that way, then all of a sudden we have a
- right triangle that's kind of going across this cube.
- This is a right angle over here.
- We know what this length is.
- This is 1/2 centimeters.
- If we could figure out the length of this orange side
- right here, this part that's going diagonally across the
- base of the cube, then we could use the Pythagorean
- theorem to figure out what the distance in question is, this
- longest diagonal of the cube.
- Now, to figure out this orange side, you just have to
- visualize the base of this cube.
- Let's look at this base.
- If we were just in the cube looking straight down at the
- base, it would look something like this.
- You'd have this side-- let me color code them; it will make
- it easier to visualize I think-- you have this side
- right here, which I will do in this blue color.
- So I'll draw it over here.
- You have this side in that blue color.
- And then you have this side over here, which I'll do in
- the magenta color.
- You have the side over there in the magenta color.
- They're both of length 1/2, so I'll draw them with roughly
- the same length.
- And then you have this orange side that goes across to the
- bottom of the base of the cube.
- And I'll do it as the same orange dotted line right
- there, just like that.
- Let me extend this side a little bit.
- And we know this is a right angle.
- This is a cube we're dealing with, so
- this is a right angle.
- We know that this side is 1/2 centimeter.
- We know that this side right over here is 1/2 centimeter.
- So if we wanted to figure out this diagonal or this length
- right here, we can just use the Pythagorean theorem.
- We know that 1/2 squared-- if we called this, I don't know,
- if we called this right here x-- we know that x squared,
- the hypotenuse squared, is equal to the the sum of the
- squares of these two guys, is equal to 1/2
- squared plus 1/2 squared.
- Or if we want to solve for x, we could say x is equal to the
- square root-- what's this going to be?
- The square root of 1/4, That's 1/2 squared, plus 1/2 squared,
- which is 1/4.
- Or, what's 1/4 plus 1/4?
- So x is equal to the square root of 1/4 plus 1/4, that's
- 2/4, or 1/2.
- Or you could write that as 1 over the square root of 2.
- Now, we could rationalize this, but I'll leave it like
- that for now.
- It'll keep things simple.
- So this distance right here is 1 over the square root of 2,
- or we could say this distance right over here is 1 over the
- square root of 2.
- So if we were to draw the right triangle where the
- hypotenuse is the length of that hole, let me draw that
- right triangle.
- So we have our hypotenuse, which is this
- long thing over here.
- Let me just draw it like this.
- That long thing over there.
- We have this side over here in yellow.
- This side right here.
- We have that side right there.
- It is of length 1/2 centimeters.
- And then we have the side we just figured out, this side
- right here, that x side.
- This right over here.
- This side over here, it is a right angle.
- We just figured out this length.
- It is 1 over the square root of 2 centimeters.
- Now, to figure out the length of the hole, we can apply the
- Pythagorean theorem again.
- This is, let me say h for hole.
- Length of hole.
- Actually, even better, let's say l.
- l for length.
- Then we can say that l squared is going to be equal to 1 over
- the square root of 2 squared-- we're just applying the
- Pythagorean theorem-- plus 1 squared.
- And once again, we know that this is the longest side.
- It's opposite the right angle.
- So it's going to be the sum of the squares of
- the other two sides.
- And so that's going to be equal to-- so we get l squared
- is equal to-- what's 1 over the square root of 2 squared?
- Well, 1 squared is 1, over the square root of 2 squared is 2.
- And then you're going to have plus 1/2 squared.
- 1/2 squared is plus 1/4.
- So l squared is equal to 1/2 plus 1/4.
- 1/2 plus 1/4 is what?
- Let's do it over here.
- 1/2-- we can write it as 2/4-- plus 1/4 is equal to 3/4.
- So this right here is equal to 3/4.
- So we have l squared-- I changed the shades of green
- just there-- is equal to 3 over 4, so l is equal to the
- square root of 3 over 4.
- So we take the square root of both sides, you get l is equal
- to the square root of 3 over the square root of 4.
- What's the square root of 4?
- The square root of 4 is 2.
- And we're done.
- The length of the hole that was drilled in this bead from
- this corner over here all the way over here is the square
- root of 3 over 2 centimeters.
- Because everything we've been dealing with has been
- centimeters.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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