3. translate the following sentence into a formula: the only sophomores uho date betty are those who like greek. (a) exhibit one term of this form (b) exhibit one atomic formula present in this formula. (c) determine the scope of each of the quantifiers that occur in this formula. (d) does this formula contain a free variable? justify (e) does this formula constitute a statement? justify. 4. verify whether the argument below is valid or not. if the argument is invalid, then write conclusion does not follow and then exhibit a combination of truth values that prove so. li, and only if, roger has entered into the contract, and the contract is legel, and roger has not performed the contract, jones will win the lausuit if roger has not accepted jones offer, roger has not entered into the contract. the fact is that roger 0 has not accepted jones' offer. therefore, jones will not win the lausuit. 10

Since opening night, attendance at play a has increased steadily, while attendance at play b first rose and then fell. equations modeling the daily attendance y at each play are shown below, where x is the number of days since opening night. on what day(s) was the attendance the same at both plays? what was the attendance? play a: y = 8x + 191 play b: y = -x^2 + 26x + 126 a. the attendance was never the same at both plays. b. the attendance was the same on day 5. the attendance was 231 at both plays on that day. c. the attendance was the same on day 13. the attendance was 295 at both plays on that day. d. the attendance was the same on days 5 and 13. the attendance at both plays on those days was 231 and 295 respectively.

1. an economy consists of three workers: larry, moe, and curly. each works 10 hours a day and can produce two services: mowing lawns and washing cars. in an hour, larry can either mow one lawn or wash one car; moe can either mow one lawn or wash two cars; and curly can either mow two lawns or wash one car. a. calculate how much of each service is produced under the following circumstances: a. all three spend all their time mowing lawns. b. all three spend all their time washing cars. c. all three spend half their time on each activity d. larry spends half of his time on each activity, while moe only washes cars and curly only mows lawns b. using your answers to part a, draw a production possibilities frontier and label points a, b, c, and d on your graph. explain why the production possibilities frontier has the shape it does. c. are any of the allocations calculated in part a inefficient? explain.

Arace is 10 kilometers long. markers will be placed at the beginning and end of the race course and at each 500 meter mark. how many markers are needed to mark the course for the race?