What is the next number in the pattern? 7,11,10,14,13
1 answer:
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Perimeter is the sum of all the sides. So we can set up an equation:
Now solve for 'x', combine like terms:
When it comes to terms with variables it's just like normal addition but we keep the variable:
So we have:
Add:
Subtract 7x to both sides:
Subtract 4 to both sides:
Divide 5 to both sides:
Now solve for 'x', combine like terms:
When it comes to terms with variables it's just like normal addition but we keep the variable:
So we have:
Add:
Subtract 7x to both sides:
Subtract 4 to both sides:
Divide 5 to both sides:
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,