A car travels 136 miles using 7 gallons of gas. At that rate, how far can the car travel using 42 gallons of gas?
2 answers:
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The cone equation gives
which means that the intersection of the cone and sphere occurs at
i.e. along the vertical cylinder of radius
when 
.
We can parameterize the spherical cap in spherical coordinates by
where
and 
, which follows from the fact that the radius of the sphere is 3 and the height at which the sphere and cone intersect is . So the angle between the vertical line through the origin and any line through the origin normal to the sphere along the cone's surface is
Now the surface area of the cap is given by the surface integral,
which means that the intersection of the cone and sphere occurs at
i.e. along the vertical cylinder of radius
We can parameterize the spherical cap in spherical coordinates by
where
Now the surface area of the cap is given by the surface integral,