Which of the following principles underlies the interaction of individual choices? a Marginal analysis is used for "how much" decisions. b People usually exploit opportunities to make themselves better off. c Resources are scarce. d There are gains from trade.

Hanna intends to give her granddaughter, melodee, her antique hat pin. this heirloom has been kept under lock and key in the wall vault in the library of hanna's house in virginia. the hat pin is currently the only item in the vault. when hanna is visiting melodee in connecticut, hanna gives melodee the only key to the vault. melodee is grateful for the present and excitedly accepts. in this situation has there been a completed gift?

Consider the local telephone company, a natural monopoly. the following graph shows the monthly demand curve for phone services and the company’s marginal revenue (mr), marginal cost (mc), and average total cost (atc) curves. 0 2 4 6 8 10 12 14 16 18 20 100 90 80 70 60 50 40 30 20 10 0 price (dollars per subscription) quantity (thousands of subscriptions) d mr mc atc 8, 60 suppose that the government has decided not to regulate this industry, and the firm is free to maximize profits, without constraints. complete the first row of the following table. pricing mechanism short run long-run decision quantity price profit (subscriptions) (dollars per subscription) profit maximization marginal-cost pricing average-cost pricing suppose that the government forces the monopolist to set the price equal to marginal cost. complete the second row of the previous table. suppose that the government forces the monopolist to set the price equal to average total cost. complete the third row of the previous table. under average-cost pricing, the government will raise the price of output whenever a firm’s costs increase, and lower the price whenever a firm’s costs decrease. over time, under the average-cost pricing policy, what will the local telephone company most likely do

Describe three scenarios in which you might utilize mathematics to investigate a crime scene, accident scene, or to make decisions involving police practice. be sure to explain how math is used in police as they work through each scenario.