Find, correct to four decimal places, the length of the curve of intersection of the cylinder 16x2 + y2 = 16 and the plane x + y + z = 12.

First find a parameterization for the curve of intersection.

Given the equation of a cylinder, a natural choice for a parameterization would be one utilizing cylindrical coordinates. Here,

which suggests we could use

with , and we get from the equation of the plane,

Now use the arc length formula:

the height of the surface increases from the center out to the sides of the road

it is c!