Use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces x^2+y^2=81 and z=5x^4.

Step-by-step explanation:

We are given that two surfaces

and

We have to find the parametrize the intersection of the given surfaces using cost and sin t with positive coefficient.

We know that intersection curve S(t) is given by

S(t)=(x(t),y(t),z(t)),

Compare with the equation of circle

Where (h,k) is center of circle and r is the radius of the circle.

The center and radius of given circle is (0,0) and 9.

because cost  takes along x - axis and sin t takes along y- axis.

When we substitute these values then we get

()

Hence,

Substitute the value of x in second equation of surface

Hence, the intersection curve s(t) is given by

b. y = 4x - 2

step-by-step explanation:

to find a parallel equation to an equation, make sure both equations have the same slope but different y-intercepts.

y = 4x + 1

4 = slope

1 = y-intercept

y = 4x - 2

4 = slope

-2 = y-intercept

they both have the same slope but different y-intercepts, so, b. is the correct answer.

and to check your work, you can graph both equations. : )