Question

Find a vector func The cone ction, r), that represents the curve of intersection of the two surfaces. z-r +y2 and the plane z 8 +y a. r()-i+ (0.03125t-8)j+ (0.03125t +8)k b. r(t)-i+ (0.0625t +16)j+ (0.0625t2 +16)k c. r r) i (0.0625t-16)j+(0.0625t2 +16)k d. r()-i+ (0.0625t-16)j+ (0.0625f2 +16)k

Answer 1

first note that if it is cone then must be

z=√(x²+y²)

↓↓↓

the way to do this problem is to  use Cartesian coordinates to get the relations between the variables, then to translate that to a vector function.

here in this case we have two equations for z

One is

z²=x²+y² and z²=(8+y)²=y²+16y+64

equate them we obtian

x²+y²=y²+16y+64

=> 16y=x²-64

=> y= (x²-64)/16

now use parameter t we have

r(t) =t i + (t²-64)/16 j + (t²+64)/16 k