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## Homework Statement

A cup is represented by the surface -(z-1)

^{2}+ x

^{2}+ y

^{2}= 1

and it is on a table represented by the plane z=0

a) find the angle at which the cup intersects the table

b) find the equation of the normal line to the cup at the point (0, √2 , 2)

c) find the equation the tangent plane at the point (0, √2 , 2)

## Homework Equations

## The Attempt at a Solution

for a) I use cos θ = a⋅b / |a||b|

a and b are normals of the cup and table at the point of intersect, which are found by finding the gradients.

∇ƒ

_{cup}= <Fx, Fy, Fz> = < 2x , 2y, -2(z-1) >

∇ƒ

_{table}= <Fx, Fy, Fz> = < 0 , 0 , 1 >

so at the point of contact between the cup and table, z is 0.

plugging that into the equation of the surface of the cup :

-(0-1)

^{2}+ x

^{2}+ y

^{2}= 1

x

^{2}+ y

^{2}= 2

so the intersection between the cup and table is a circle with radius √2

then I find a point on the circle x

^{2}+ y

^{2}= 2 , setting x = 0 then y = √2

then a point is (0,√2, 0)

plugging this point into < 2x , 2y, -2(z-1) > and < 0 , 0 , 1 >:

< 0, 2√2, 2 > and < 0 , 0 , 1 >

∴ cos θ = a⋅b / |a||b|

b) plugging the point (0,√2, 2) into < 2x , 2y, -2(z-1) >:

< 0, 2√2, -2 >

equation of the normal line to the cup at point (0,√2, 2) =

(0,√2, 2) + t < 0, 2√2, -2 > = ( 0 , √2 + 2√2 t , 2-2t )

c) equation of tangent plane at point (0,√2, 2) :

2√2(y- √2) -2(z-2) = 0

can someone check my work?

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