The water-supply manager for a Midwestern city needs to supply the city with at least 10 million gallons of potable (drinkable) water per day. The supply may be drawn from the local reservoir or from a pipeline to an adjacent town. The local reservoir has a maximum daily yield of 5 million gallons of potable water, and the pipeline has a maximum daily yield of 10 million gallons. By contract, the pipeline is required to supply a minimum of 6 million gallons/day. If the cost for 1 million gallons of reservoir water is $\$ 300$ and that for pipeline water is $\$ 500$, how much water should the manager get from each source to minimize daily water costs for the city?

Use the Laplace transform to solve the given initial-value problem.

$$y ^ { \prime \prime } - 2 y ^ { \prime } - 8 y = 5 , \quad y ( 0 ) = 1 , \quad y ^ { \prime } ( 0 ) = 0.$$

Use the given transformation to evaluate the integral. double integral (x-3y)dA, where R is the triangular region with vertices (0, 0), (2, 1) and (1, 2); x=2u+v, y=u+2v

The graph models the number of people N(t) who have come down with the flu t weeks after its initial outbreak in a town with a population of 50,000 citizens. a. Describe what N'(t) represents and how it behaves as t increases. b. What does the derivative tell us about how this town is affected by the flu outbreak?

Soundex produces two models of satellite radios. Model A requires 15 min of work on Assembly Line I and 10 min of work on Assembly Line II. Model B requires $10 \mathrm{~min}$ of work on Assembly Line I and $12 \mathrm{~min}$ of work on Assembly Line II. At most $25 \mathrm{hr}$ of assembly time on Line I and $22 \mathrm{hr}$ of assembly time on Line II are available each day. Soundex anticipates a profit of $\$ 12$ on model $A$ and $\$ 10$ on model B. Because of previous overproduction, management decides to limit the production of model A satellite radios to no more than 80/day.

a. To maximize Soundex's profit, how many satellite radios of each model should be produced each day?

b. Find the range of values that the contribution to the profit of a model A satellite radio can assume without changing the optimal solution.

c. Find the range of values that the resource associated with the time constraint on Assembly Line I can assume.

d. Find the shadow price for the resource associated with the time constraint on Assembly Line I.

e. Identify the binding and nonbinding constraints.

The ordered pairs below give the average water temperatures C (in degrees Celsius) at several depths d (in meters) in the Indian Ocean.

$$\begin{array}{ll} (1000,4.85) & (2500,1.888) \\ (1500,3.525) & (3000,1.583) \\ (2000,2.468) & (3500,1.422) \end{array}$$

Find k for each pair of coordinates. Then find the mean value of k to find the constant of proportionality for the model you chose in part (b).

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type A vessel has 60 deluxe cabins and 160 standard cabins, whereas a type B vessel has 80 deluxe cabins and 120 standard cabins. Under a charter agreement with the Odyssey Travel Agency, Deluxe River Cruises is to provide Odyssey with a minimum of 360 deluxe and 680 standard cabins for their 15-day cruise in May. It costs $\$ 44,000$ to operate a type A vessel and $\$ 54,000$ to operate a type B vessel for that period.

a. How many of each type of vessel should be used to keep the operating costs to a minimum?

b. Find the range of values that the cost of operating a type A vessel can assume without changing the optimal solution.

c. Find the range of values that the requirement for deluxe cabins can assume.

d. Find the shadow price for the requirement for deluxe cabins.

In the following exercise, solve each linear programming problem by the method of corners.

$$\begin{array}{lr} \text { Maximize } & P=2 x+5 y \\ \text { subject to } & 2 x+y \leq 16 \\ & 2 x+3 y \leq 24 \\ & y \leq 6 \\ & x \geq 0, y \geq 0 \end{array}$$

In the following exercise, use the method of corners to solve the linear programming problem.

Find the maximum and minimum of $Q=2 x+5 y$ subject to

$$\begin{array}{r} x+y \geq 4 \\ -x+y \leq 6 \\ x+3 y \leq 30 \\ x \leq 12 \\ x \geq 0, y \geq 0 \end{array}$$

A farmer uses two types of fertilizers. A 50-lb bag of Fertilizer A contains 8 lb of nitrogen, 2 lb of phosphorus, and 4 lb of potassium. A 50-lb bag of Fertilizer B contains $5 \mathrm{lb}$ each of nitrogen, phosphorus, and potassium. The minimum requirements for a field are $440 \mathrm{lb}$ of nitrogen, $260 \mathrm{lb}$ of phosphorus, and $360 \mathrm{lb}$ of potassium. If a 50-lb bag of Fertilizer $A$ costs $\$ 30$ and a 50-lb bag of Fertilizer $B$ costs $\$ 20$, find the amount of each type of fertilizer the farmer should use to minimize his cost while still meeting the minimum requirements.

Consider the following function f(x) = ln(1 + 2x), a = 3, n = 3, 2.6 ≤ x ≤ 3.4 Approximate f by a Taylor polynomial with degree n at the number a. Use Taylor's Inequality to estimate the accuracy of the approximation f(x) ≈ Tn(x) when x lies in the given interval. (Round your answer to six decimal places.) |R3(x)| ≤ 0.000001

Your weekly cost (in dollars) to manufacture x bicycles and y tricycles is C(x, y)=24,000+60x+20y. a. What is the marginal cost of a bicycle? Of a tricycle? b. Describe the graph of the cost function C. c. Describe the slice by y=100. What cost function does this slice describe? d. Describe the level curve z=72,000. What does this curve tell you about costs?

A simple economy consists of two industries: agriculture and manufacturing. The production of $1$ unit of agricultural products requires the consumption of $0.2$ unit of agricultural products and $0.3$ unit of manufactured goods. The production of $1$ unit of manufactured goods requires the consumption of $0.4$ unit of agricultural products and $0.3$ unit of manufactured goods.

a. Find the total output of goods needed to satisfy a consumer demand for $100$ million dollars worth of agricultural products and $150$ million dollars worth of manufactured goods.

b. Find the value of the goods consumed in the internal process of production in order to meet the gross output.

At a certain university, three bookstores - the University Bookstore, the Campus Bookstore, and the Book Mart-currently serve the university community. A survey conducted at the beginning of the fall quarter found that the University Bookstore and the Campus Bookstore each had $40 \%$ of the market, whereas the Book Mart had $20 \%$ of the market. Each quarter, the University Bookstore retains $80 \%$ of its customers but loses $10 \%$ to the Campus Bookstore and $10 \%$ to the Book Mart. The Campus Bookstore retains $75 \%$ of its customers but loses $10 \%$ to the University Bookstore and $15 \%$ to the Book Mart. The Book Mart retains $90 \%$ of its customers but loses $5 \%$ to the University Bookstore and $5 \%$ to the Campus Bookstore. Using the data, determine the expected market share that each store will have at the beginning of the fourth quarter.