Jessica has $50 to spend on new clothes.
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Use Stokes' theorem for both parts, which equates the surface integral of the curl to the line integral along the surface's boundary.
a. The boundary of the hemisphere is the circle in the plane
, where the curl is
. Green's theorem applies here, so that
which means the value of the line integral is 3 times the area of the circle, or .
b. The closed sphere has no boundary, so by Stokes' theorem the integral is 0.
Given:
The leading coefficient of a polynomial is 8.
Polynomial roots are 1 and 2.
The graph passes through the point (4,5).
To find:
The 3rd root and the equation of the polynomial.
Solution:
The factor form of a polynomial is:
Where, a is a constant and are the roots of the polynomial.
Polynomial roots are 1 and 2. So, and
are the factors of the polynomial.
Let the third root of the polynomial by c, then is a factor of the polynomial.
The leading coefficient of a polynomial is 8. So, a=8 and the equation of the polynomial is:
The graph passes through the point (4,5). Putting , we get
Divide both sides by 48.
Therefore, the 3rd root on the polynomial is .