The method of Problem 20 can be extended to second order equations with variable coefficients. If y1 is a known nonvanishing solution of y′′ + p(t)y′ + q(t)y = 0, show that a second solution y2 satisfies (y2 /y1 )′ = W (y1 , y2 )/y21 , where W (y1 , y2 ) is the Wronskian of y1 and y2 . Then use Abel’s formula [Eq. (23) of Section 3.2] to determine y2 .

Find the area of the triangle shown 3 by 6

Martha has a deck of cards. she has lost some of the cards, and now the deck only contains nine spades, eleven diamonds, eight clubs, and twelve hearts. martha predicts that whenever she draws a card from the deck without looking, she will draw a club one-fifth of the time. which activity would best allow martha to test her prediction? a. randomly draw a card from the box and see if it is a club. b. randomly draw a card. then, continue to draw another card until all eight clubs are drawn. c. randomly draw and replace a card 120 times. then, observe how close to 30 times a club is drawn. d. randomly draw and replace a card 100 times. then, observe how close to 20 times a club is drawn.

Acircle has a circumference of 7.850 units. what is its radius?