A rectangle is constructed with its base on the diameter of a semicircle with radius 2 and with its two other vertices on the semicircle. What are the dimensions of the rectangle with maximum area?

L = 2*√2

w = √2

Step-by-step explanation:

Given:

A rectangle is constructed with its base on the diameter of a semicircle with radius 2 and with its two other vertices on the semicircle.

Find:

What are the dimensions of the rectangle with maximum area?

Solution:

- Let the length and width of the rectangle be L and w respectively.

- We know that Length L lie on the diameter base. So , L < 4 and the width w is less than 2 . w < 2.

- Using the Pythagorean Theorem, we relate the L with w using the radius r = 2 of the semicircle.

                           r^2 = (L/2)^2 + (w)^2

                           sqrt (4 - w^2 ) = L / 2

                           L = 2*sqrt (4 - w^2 )           L < 4 , w < 2

- The relation derived above is the constraint equation and the function is Area A which is function of both L and w as follows:

                          A ( L , w ) = L*w

- We substitute the constraint into our function A:

                          A ( w ) = 2*w*sqrt (4 - w^2 )

- Now we will find the critical points for width w for which A'(w) = 0

                         A'(w) = 2*sqrt (4 - w^2 ) - 2*w^2 / sqrt (4 - w^2 )  

                         0 = [2*sqrt (4 - w^2 )*sqrt (4 - w^2 )   - 2*w^2] / sqrt (4 - w^2 )  

                         0 = 2*(4 - w^2 )   - 2*w^2

                         0 = -4*w^2 + 8

                         8/4 = w^2

                         w = + sqrt ( 2 )   0 < w < 2

- From constraint equation we have:

                          L = 2*sqrt (4 - 2 )

                          L = 2*sqrt(2)

step-by-step explanation:

this woman was an important writer and propagandist during the american revolution. she was a member of the committee of correspondence (in fact the committee was formed in her home) and was an advisor to many of the revolutionary leaders, including george washington and john adams. who was this revolutionary woman?