Section 2.5, 2.6 & 2.7 Worksheet. Optimization

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1 MATH 1142 Section 2.5, 2.6 & 2.7 Worksheet NAME Optimization Optimization problems are very pracitcal. They can be used in business, transportation, medicine and much more. This is the fun thing about optimization, it is extremely applicable. Now, not all of the problems we will look at will be the most practical or interesting problems, (we would have to do a lot more work for things to get more interesting), but it is important to notice that optimization is an area of mathematics that has a variety of applications and is a very valuable area of study. Optimization problems are about maximizing the things that you want more of (i.e. profits) and minimizing the things that you would like less of (i.e. costs). Luckily, in the last few sections we have discussed how to find the maximums and minimums of particular functions. 1. Suppose that a ball is thrown straight up into the air and its height after t seconds is t 16t 2. Determine how long it will take for the ball to reach its maximum height and determine the maximum height. 2. Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible. 1

2 3. A rectangular storage container with an open top is to have volume of 10 m 3. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container. 4. The rate (in mg carbon/m 3 /h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function P = 100l l 2 + l + 4 where l is the light intensity (measured in thousands of foot-candles). For what light intensity is the rate of photosynthesis a maximum? 2

3 5. A boat leaves a dock at 2:00 pm and travels due south at the speed of 20 km/h. Another boat has been heading due eats at 15 km/h and reaches the same dock at 3:00 pm. At what time were the two boats closest together? Inventory Problems When a firm regularly orders and stores supplies for later use or resale it must decide on the size of each order. If it orders enough supplies to last an entire year, the business will incur heavy carrying costs (i.e. insurance, storage, the cost of the capital ties up in inventory, etc). To reduce these carrying costs the firm could order small quantities of the supplies at regular intervals. However, such a policy increases ordering costs (i.e. freight/delivery costs, clerical costs, etc). The problem lies in balancing these two extremes 6. Supermarket manager anticipates that 1200 cases of orange juice will be sold at a steady rate during the next year. The manager plans on ordering these cases by placing equally spaced orders the same size throughout the year. Given that it costs $8 to carry one case of orange juice in inventory for 1 year, find the carrying cost of if (a) The manager places only one order during the year. (b) The manager places two orders during the year 3

4 (c) The manager places four orders during the year. Now suppose that the manager wants to establish an optimal inventory policy for frozen orange juice. Again, it is estimated that the total of 1200 cases will be sold at a steady rate during the next year. Use the following data to determine the economic order quantity; that is the order size that minimizes the total ordering and carrying cost: The ordering cost for each delivery is $75. It costs $8 to carry one case of orange juice in inventory for one year. 4

5 Business/Economics It may be important to know the following background information about economics. Definitions The cost function is the cost of producing x units of a certain product and is denoted C(x). The marginal cost is the rate of change of C with respect to x, C (x). The demand (or price) function, p(x), is the price per unit that the company can charge if it sells x units. The revenue function, R(x), is the total revenue and is modeled by xp(x) if x units are sold and the price per unit is p(x). The marginal revenue function is the rate of change of revenue with respect to the number of units sold, R (x). The profit function, P (x) when x units are sold is modeled by P (x) = R(x) C(x). The marginal profit function, P (x), is the derivative of the profit function. 7. A baseball team plays in a stadium that holds 55, 000 spectators. With ticket prices at $10, the average attendance has been 27, 000. When ticket prices were lowered to $8, the average attendance rose to 33, 000. a. Find the demand function, assuming that it is linear. b. How should ticket prices be set to maximize revenue? 5

6 8. Suppose that the demand equation for a monopolist is P (x) = x and the cost function C(x) = 50x + 10, 000. Find the value of x that maximizes the profit and determine the corresponding price and total profit for this level of production. 6