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有理式的應用 3 (英): 有理式的應用 3
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- Two different hoses are being used to fill a fish pond.
- Used together, the two hoses take 12
- minutes to fill the pond.
- If used alone, one hose is able to fill the pond 10
- minutes faster than the other.
- So 10 minutes faster than the other hose.
- How long does each hose take to fill the pond by itself?
- So let's think about each of the hoses.
- We have a faster hose, and then we have a slower hose.
- And let's say that the faster hose fills the pond.
- Let's say it takes him f minutes per pond.
- Now how long is it going to take the slower hose?
- Well, the faster hose does it in 10 minutes less.
- It's 10 minutes faster.
- So the slower hose is going to take 10 minutes more.
- So the slower hose is going to take f plus 10 minutes per
- fish pond or per pond.
- Now this is in minutes per pond, but if we want to be
- able to add rates together, we should really think about in
- terms of ponds per minute.
- So let's rewrite each of these statements
- as ponds per minute.
- You could write this as f minutes per 1 pond or f plus
- 10 minutes per 1 pond.
- And if you just take the inverse of each of these
- statements, these ratios are equivalent to saying 1 pond
- per f minutes.
- So it's really not saying anything else.
- I'm just inverting the ratio.
- Or you could think of it as 1/f ponds per minute.
- Same logic right here.
- We could essentially rewrite this ratio as 1 over f plus 10
- ponds per minute.
- So now we have the rate of the faster hose.
- We have the rate of the slower hose.
- How many pounds per minute for the faster hose?
- How many pounds per minute for the slower hose?
- If we add these two rates, we'll know the ponds per
- minute when they're acting together.
- So if we have 1/f ponds per minute plus 1 over f plus 10
- ponds per minute.
- This is the faster hose, this is the slower hose.
- This'll tell us how many ponds for
- minute they can do together.
- Now, we know that information.
- They say the two hoses take 12 minutes.
- So let me write that over here.
- So combined.
- The combined take 12 minutes per pond.
- So what is their combined rate in terms of ponds per minute?
- So you could view this as 12 minutes per one pond.
- You could take the inverse of this or take the ratio in
- terms of ponds per minute instead and
- you get 1/12 minutes.
- Sorry, 1/12 ponds per minute.
- In one minute combined, they'll fill 1/12 of a pond.
- Which makes complete sense.
- Because it takes them 12 minutes to
- fill the whole thing.
- So in one minute they'll only do 1/12 of it.
- So this is their combined rate in ponds per minute.
- This is also their combined rate in ponds per minute.
- So this is going to be equal to 1/12.
- And now we just have to solve for f and then f plus 10 is
- going to be how long it takes the slower hose.
- So let's multiply.
- Let's see what we could do.
- We could multiply both sides of this equation times f and
- times f plus 10.
- So let's do that.
- So I'm going to multiply both sides of this equation times f
- and f plus 10 times both sides of this equation.
- So f and f plus 10.
- Scroll down a little bit.
- Scroll to the left, so we have some real estate.
- So let's distribute this f times f plus 10.
- So if we multiply f times f plus 10 times 1/f, that f and
- that f will cancel out and we're just going to be left
- with an f plus 10.
- That's when you multiply that term times the f
- times f plus 10.
- Now, when you multiply this term, when you multiply 1 over
- f plus 10 times f times f plus 10, this and this will cancel
- out and you're just left with an f.
- So you have plus f is equal to-- and you have over 12.
- Actually, well in a second, let me multiply all sides of
- this equation by 12.
- I'll do that next in a second.
- So let's just say this is going to be equal to 1/12
- times f squared.
- f times f is f squared.
- Plus 10f.
- And now let's just multiply both sides of
- this equation by 12.
- I could've done it in the last step so that we don't have any
- fractions here.
- And so the left-hand side we get 12 times f.
- We get 12f plus 120 plus 12f.
- The right-hand side, that and that cancels out and you are
- left with f squared plus 10f.
- And now we have a quadratic.
- We just have to get it into a form that we know how to
- manipulate or deal with.
- Before that, we can simplify it.
- We have a 12f and a 12f.
- So this becomes 24f plus 120 is equal to f
- squared plus 10f.
- And then let's get all of this stuff out of
- the left-hand side.
- They'll get all on the right-hand side.
- So from both sides of this equation, let's subtract 24f
- and a negative 120 or minus 120.
- So you have minus 24f minus 120.
- The left-hand side just becomes 0.
- That was the whole point.
- Right-hand side is f squared.
- 10 minus 24f is negative 14f.
- Minus 120.
- Now we could factor this.
- Let's see, if you do 20 timea-- yeah, that looks like
- it would work.
- So negative 20 and 6, when you take their product, give you
- negative 120.
- And negative 20 plus 6 is negative 14.
- So we could factor this right-hand side as 0 is equal
- to f minus 20 times f plus 6.
- When you multiply negative 20 times 6, you get negative 120.
- Negative 20 plus 6 is negative 14.
- And the only way that that's going to be equal to 0 is if f
- minus 20 is equal to 0 or f plus 6 is equal to 0.
- Add 20 to both sides of this equation.
- You get f is equal to 20.
- Remember, f is how many minutes does it take for the
- fast hose to fill the pond.
- And then, if you take this one, you
- subtract 6 from both sides.
- You get f is equal to negative 6.
- Now, when we're talking about how many minutes does it take
- for the fast hose to fill the pond, it doesn't make any
- sense to say that it takes it negative 6
- minutes to fill the pond.
- So we can't use this answer.
- We need a positive answer.
- So this is how many minutes it takes the fast hose to fill
- the pond. f is equal to 20.
- So this right here, the faster hose takes
- 20 minutes per pond.
- I'll write it here.
- 20 minutes per pond is the fast hose.
- And then the slower hose, it takes 10 minutes more.
- It's f plus 10.
- So it takes 30 minutes per pond.
- And we're done.
- I don't want to confuse you with this stuff.
- The faster hose takes 20 minutes.
- Slower hose takes 30 minutes per pond.
- If they were to do it together, it would take 12
- minutes, which is a little bit more than half.
- If you had two faster hoses, it would take 10 minutes.
- But this guy's a little bit slower, so it's taking you a
- little bit more than 10 minutes.
- So it makes sense.
- It takes you 12 minutes when they're working together.
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