Country Workshop manufactures both finished and unfinished furniture for the home. The estimated quantities demanded each week of its rolltop desks in the finished and unfinished versions are x and y units when the corresponding unit prices in dollars, p and q, respectively, are described in the following equations. p = 200 - 1/5 x - 1/10 yq = 160 - 1/10 x - 1/4 yFor the weekly total revenue function R(x, y), compute the values of R listed below using the given unit prices. R(80, 40) = $R(60, 100) = $

Given that y varies directly to the square root of (x + 1), and that y = 1 when x = 8,(a)express y in terms of .x,(b)find the value of y when x = 3,(c)find the value of x when y = 5.

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.

What are the measures of angles a, b, and c? show your work and explain your answers. 25 points