Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces
Parameter

Question

Quizplus · Accepted Answer

To find the intersection of the given surfaces, we need to set them equal to each other and solve for x, y, and z in terms of the given parameter. 
Setting the given surfaces equal to each other, we have: 
x^2 + y^2 + z^2 = 16 
x + y + z = t

Using the method of Lagrange multipliers, we can find that the solution is

x = (2+t)/3, y = (2+t)/3, z = (10-2t)/3

Therefore, the vector-valued function representing the intersection of the given surfaces is:

r(t) = ((2+t)/3)i + ((2+t)/3)j + ((10-2t)/3)k

Simplifying, we get 
r(t) = (2/3)i + (2/3)j + (10/3)k + (t/3)i + (t/3)j - (2/3)t*k

which matches the coordinates of choice D.

Find a Vector-Valued Function, Using the Given Parameter, to Represent