Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem.

For the rule of 78, for a 12-month period, the last term in the sequence is 12 and the series sums to 78. for an 10 month period, the last term is and the series sum is . for a 15 month period, the last term is and the series sum is . for a 20 month period, the last term is and the series sum is

Asquare picture with sides of 32 inches is to be mounted on a square mat with sides of 52 inches. find the area of the mat that will remain uncovered by the picture.

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.