Question

The position of a bug is described by the parametric equation (x, y) = (2 − 12t, 1 + 5t). Explain why the speed of the bug is 13 cm/sec. Change the equation to obtain the description of a bug moving along the same line with speed 26 cm/second.

Answer 1

@zepdrix @agent0smith @skullpatrol

Answer 2

Notice...
x = 2 - 12t <= in one second, it advances 12 units in the negative x direction
y = 1 + 5t <= in one second, it advances 5 units in the positive y direction

Answer 3

OH. It creates 5-12-13 triangle!

Answer 4

Now to have the speed be 26cm/sec do I need to double 5 and 12?

Answer 5

Yes. Changing those shouldn't affect the line at all... I don't think. You might have to double everything in both equations. We can check:

x = 2 - 12t <==== solve for t
y = 1 + 5t <==== plug in here

x = 4 - 24t <==== solve for t
y = 2 + 10t <==== plug in here

Do you get the same equation? If not, then it's not moving along the same line. If that fails, just try doubling only the 12 and 5 and see if that yields the same line as the original.

Answer 6

OK I am trying it out right now

Answer 7

Well the slope is basically the same (if you completely simplify it). But do you think that I should make a right triangle and solve for the hypotenuse and see if it is 26?

Answer 8

Just checked for the hypotenuse and it is 26.

Answer 9

@agent0smith Can you help me with another question please?

Answer 10

But you need to check it's the exact same line...

Answer 11

So you are saying to solve for t and then plug it back in to the parametric equation for the y. How would I solve for t? (This might be a dumb question)

Answer 12

You should know how to solve for t. Get it alone by doing math.

Answer 13

I am so sorry but I am really bad with parametric equations. But if I am wrong with this please correct me.

So would the x value be like 0 in the coordinate (0,5)

Answer 14

x = 2 - 12t rearrange to solve for t

Answer 15

Or...
2-12(1) = 2 - 12 = -10 <- x
1 + 5(1) = 1 + 5 = 6 <- y

Answer 16

No that doesn't make any sense.

Answer 17

What do you mean by "rearrange to solve for t"?

Answer 18

Get t alone. so t = something.

Answer 19

2-12t = x
-12t = x - 2
t = (x/(-12)) + 1/6

Answer 20

Is that right?

Answer 21

Yes. Now you plug that into the y.

Answer 22

y = 1 + 5((x/(-12)) + 1/6)
y = 1 + (5x/(-12) + 5/6)
y = (1 5/6) + (5x/(-12))

Answer 23

Now you gotta make sure the second gives the same line.

Answer 24

ok

Answer 25

4 - 24t = x
-24t = x - 4
t = (x/(-24) + 1/6)

y = 2 + 10(x/(-24) + 1/6)
y = (2 1/6) + (5x/(-12))

Answer 26

2 - 24t = x
1 + 10t = y

-24t = x - 2
t = (x/(-24) + 1/12)

1 + 10((x/(-24) + 1/12))
1 + 10x/(-24) + 10/12
1 + 5x/(-12) + 5/6
1 5/6 + (5x - 12) <- this works!

Answer 27

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Answer 28

Can I ask you like two more questions please?

Answer 29

AND THANK YOU!

Answer 30

I gotta go, i'll be back later.

Answer 31

Ok thank you for your help though :)