Simplify each of the following expressions by applying one of the theorems. state the theorem used. (a) x′y′z + (x′y′z)′ (b) (ab′ + cd)(b′e + cd) (c) acf + ac′f (d) a(c + d′b) + a′ (e) (a′b + c + d)(a′b + d) (f) (a + bc) + (de + f )(a + bc)′

Which statement explains how you could use coordinate geometry to prove the opposite sides of a quadrilateral are congruent? a. use the slope formula to prove the slopes of the opposite sides are the same. b. use the slope formula to prove the slopes of the opposite sides are opposite reciprocals. c. use the distance formula to prove the lengths of the opposite sides are the same. d. use the distance formula to prove the midpoints of the opposite sides are the same.

(20 points) a bank has been receiving complaints from real estate agents that their customers have been waiting too long for mortgage confirmations. the bank prides itself on its mortgage application process and decides to investigate the claims. the bank manager takes a random sample of 20 customers whose mortgage applications have been processed in the last 6 months and finds the following wait times (in days): 5, 7, 22, 4, 12, 9, 9, 14, 3, 6, 5, 15, 10, 17, 12, 10, 9, 4, 10, 13 assume that the random variable x measures the number of days a customer waits for mortgage processing at this bank, and assume that x is normally distributed. 2a. find the sample mean of this data (x ě…). 2b. find the sample variance of x. find the variance of x ě…. for (c), (d), and (e), use the appropriate t-distribution 2c. find the 90% confidence interval for the population mean (îľ). 2d. test the hypothesis that îľ is equal to 7 at the 95% confidence level. (should you do a one-tailed or two-tailed test here? ) 2e. what is the approximate p-value of this hypothesis?

Judy garland electronics operate on a net-profile rate of 20% on its printer cables. if the markup is $8.95 and the overhead is $4.31,find the net profit and the selling price? use the net-profit rate formula to solve this problem.