Question

how do you do this
If the height of the cone is 5 and the base radius is 4, write a parameterization of the cone in terms of r=s and θ=t.
x(s,t)= ,
y(s,t)= , and
z(s,t)= , with
≤s≤ and
≤t≤ .

Answer 1

@wio

Answer 2

Like this?

Answer 3

Alright.

Answer 4

Okay so first lets do \(z\) cause it is easiest. We want to convert the radius to height.

Answer 5

We have two points: \((s,z)=(0,5)\) and \((s,z)=(4,0)\)

Answer 6

The equation of a line that goes through the points \((x_1,y_1)\) and \((x_2,y_2)\) is given by: \[
y-y_1 = \frac{y_2-y_1}{x_2-x_1} (x-x_1)
\]

Answer 7

Use this if formation to find \(z(s)\)

Answer 8

\(z(s,t)=z(s)\) since \(z\) will be constant with respect to \(t\).

Answer 9

We also know that \(0\le s \le 4\)

Answer 10

@dan815 You following me?

Answer 11

ya im reading

Answer 12

i dont understand that equation of a linne

Answer 13

will i get the same answer with y = mx+b

Answer 14

oh nvm now i see go on

Answer 15

Basically \(z(s,t)\) is a line. \(z(s,t)=ms+b\)

Answer 16

So can you find \(z\) to start with?

Answer 17

so (5/-4)x + 5?

Answer 18

yes, but not \(x\), it's \(s\)

Answer 19

right

Answer 20

do you remember the parametrization of a circle?

Answer 21

x^2+y^2 = 16?

Answer 22

sintheta^2+cos^2

Answer 23

sin(t)^2+cos(t)^2=16 to sin(t)^2+cos(t)^2=0

Answer 24

It's \((x(t), y(t)) = (\cos(t),\sin(t))\)

Answer 25

how do u write an equation where that circle is decreasing as z increases

Answer 26

But this is for the unit circle, if you want it to have radius \(s\), we multiply that in: \[
x(s,t)= s\cos(t)\\
y(x,t)=s\sin(t)
\]

Answer 27

\[
x(s,t)= s\cos(t)\\
y(s,t)=s\sin(t)
\]

Answer 28

why does that work

Answer 29

Remember the \(z\) changes is because \(s\) that is increasing.

Answer 30

It works because of the equation of a circle: \[
x^2+y^2= r^2 \\
[s\cos(t)]^2+[s\sin(t)]^2=r^2\\

s^2\cos^2(t)+s^2\sin^2(t)=r^2\\
s^2(\cos^2(t)+\sin^2(t))=r^2\\
s^2= r^2
\]

Answer 31

ok gotcha

Answer 32

thanks

Answer 33

Remember the bounds of \(s\) and \(t\)

Answer 34

0 - 4 and 0-2pi

Answer 35

good

Answer 36

ty teacher :D