Math 1051-20 FINITE MATHEMATICS FOR THE SOCIAL AND MANAGEMENT SCIENCES Summer 2015 CRN: 30301 HOMEWORK III DUE Monday, August 3, at the beginning of the lecture 1. Solve the following Linear Programming Problem geometrically: Minimize the function x 4y subject to the constraints: x y 2 x + y 8 x 3y 0 19x 40y 38 x 0, y 0 Draw a large (at least half a page), neat diagram of the feasible region, with all lines and corner points clearly labelled, and the value of the objective function at each corner point clearly denoted ... and end by noting which point solves the LPP. 2. A city council is planning to build several hotels along its beach-front. They will be of two types: a convention-style hotel with 400 rooms costing $75 million to build and a vacation-style hotel with 200 rooms costing $40 million. The city council wants a total capacity of at least 3,000 rooms, wants there to be no more than 15 hotels altogether, with at most 8 conventionstyle, and with no more that twice as many convention-style hotels as vacation-style. How many hotels of each type should the council request in order to minimize cost? (a) Depict the 2-D LPP equivalent to the above 'real-world' problem. Clearly state what each of your variables denotes. (b) Solve the 2-D LPP geometrically. Draw a large neat diagram of the feasible region, with all lines and corner points clearly labelled, and the value of the objective function at each corner point clearly denoted. (c) Come back out and answer the question at the end of the 'real-world' problem. 3. I form a 3-digit number at random using digits chosen from 1 to 9, allowing repeats; determine the probability that my number ... (a) begins and ends with 1. (b) has 1 as the middle digit. (c) has all three digits the same. (d) has exactly one repeated digit (i.e. two digits the same and one dierent). Work the probabilities out completely, leaving them as fractions in their simplest form. 4. A bag contains 6 marbles: 1 red, 2 blue, and 3 green; I reach into the bag and pick 4 marbles (simultaneously) at random. Determine the probability that my pick has ... (a) at most 2 green marbles. (b) at least 2 green marbles. (c) all three colors represented. (d) at most two colors represented. Work the probabilities out completely, leaving them as fractions in their simplest form. continued ... 5. I create a 7-digit password by randomly mixing 3 numbers (chosen from 1 to 9, allowing repeats) and 4 letters (chosen from A to Z, allowing repeats). Determine the probability that my password has ... (a) 4 A's. (b) At least 2 A's. (c) No two numbers next to each other. (d) The three numbers next to each other. m m You may leave your probabilities in terms of mn , Pn , Cn , and factorials. *************************************** x line1 y 2 0 10 0 -2 8 8 0 0 12 2 0 10 0 8 0 4 0 -0.95 3.8 line1 line 2 line 3 line 4 10 line 2 line 3 8 line 4 6 y 4 2 0 0 2 4 6 8 -2 -4 x 10