Demand function. Given the demand function

$$p=D(x)=\frac{1,000}{x}$$

find the average price (in dollars) over the demand interval $[400, 600]$.

Seneca Hill Winery recently purchased land for the purpose of establishing a new vineyard. Management is considering two varieties of white grapes for the new vineyard: Chardonnay and Riesling. The Chardonnay grapes would be used to produce a dry Chardonnay wine, and the Riesling grapes would be used to produce a semi-dry Riesling wine. It takes approximately four years from the time of planting before new grapes can be harvested. This length of time creates a great deal of uncertainty concerning future demand and makes the decision concerning the type of grapes to plant difficult. Three possibilities are being considered: Chardonnay grapes only; Riesling grapes only; and both Chardonnay and Riesling grapes. Seneca management decided that for planning purposes it would be adequate to consider only two demand possibilities for each type of wine: strong or weak. With two possibilities for each type of wine it was necessary to assess four probabilities. With the help of some forecasts in industry publications management made the following probability assessments.

$$\begin{matrix} \text{ } & \text{Riesling} & \text{Demand}\\ \text{Chardonnay Demand} & \text{Weak} & \text{Strong}\\ \text{Weak} & \text{.05} & \text{.50}\\ \text{Strong} & \text{.25} & \text{.20}\\ \end{matrix}$$

Revenue projections show an annual contribution to profit of $20,000 if Seneca Hill only plants Chardonnay grapes and demand is weak for Chardonnay wine, and$70,000 if they only plant Chardonnay grapes and demand is strong for Chardonnay wine. If they only plant Riesling grapes, the annual profit projection is $25,000 if demand is weak for Riesling grapes and$45,000 if demand is strong for Riesling grapes. If Seneca plants both types of grapes, the annual profit projections are shown in the following table.

$$\begin{matrix} \text{ } & \text{Riesling} & \text{Demand}\\ \text{Chardonnay Demand} & \text{Weak} & \text{Strong}\\ \text{Weak} & \text{\$22,000} & \text{\$40,000}\\ \text{Strong} & \text{\$26,000} & \text{\$60,000}\\ \end{matrix}$$

a. What is the decision to be made, what is the chance event, and what is the consequence? Identify the alternatives for the decisions and the possible outcomes for the chance events. b. Develop a decision tree. c. Use the expected value approach to recommend which alternative Seneca Hill Winery should follow in order to maximize expected annual profit. d. Suppose management is concerned about the probability assessments when demand for Chardonnay wine is strong. Some believe it is likely for Riesling demand to also be strong in this case. Suppose the probability of strong demand for Chardonnay and weak demand for Riesling is .05 and that the probability of strong demand for Chardonnay and strong demand for Riesling is .40. How does this change the recommended decision? Assume that the probabilities when Chardonnay demand is weak are still .05 and .50. e. Other members of the management team expect the Chardonnay market to become saturated at some point in the future, causing a fall in prices. Suppose that the annual profit projections fall to $50,000 when demand for Chardonnay is strong and Chardonnay grapes only are planted. Using the original probability assessments, determine how this change would affect the optimal decision.

"Myrtle Air Express decided to offer direct service from Cleveland to Myrtle Beach. Management must decide between a full-price service using the company’s new fleet of jet aircraft and a discount service using smaller-capacity commuter planes. It is clear that the best choice depends on the market reaction to the service Myrtle Air offers. Management developed estimates of the contribution to profit for each type of service based on two possible levels of demand for service to Myrtle Beach: strong and weak. The following table shows the estimated quarterly profits (in thousands of dollars):

$$\begin{aligned} &\text { Demand for Service }\\ &\begin{array}{lcr} \text { Service } & \text { Strong } & \text { Weak } \\ \text { Full Price } & \$ 960 & -\$ 490 \\ \text { Discount } & \$ 670 & \$ 320 \end{array} \end{aligned}$$

Suppose that management of Myrtle Air Express believes that the probability of strong demand is 0.7 and the probability of weak demand is 0.3. Use the expected value approach to determine an optimal decision."

Solve the following exercise.

Find the demand function q = D(x) if the elasticity of demand is E(x) = x.

Suppose management is concerned about the probability assessments when demand for Chardonnay wine is strong. Some believe it is likely for Riesling demand to also be strong in this case. Suppose that the probability of strong demand for Chardonnay and weak demand for Riesling is 0.05 and that the probability of strong demand for Chardonnay and strong demand for Riesling is 0.40 . How does this change the recommended decision? Assume that the probabilities when Chardonnay demand is weak are still 0.05 and 0.50 .

What is a demand schedule? What is a demand curve?

The weekly demand and supply functions for Sportsman $5 \times 7$ tents are given by

$$\begin{array}{l} p=-0.1 x^{2}-x+40 \\ p=0.1 x^{2}+2 x+20 \end{array}$$

respectively, where p is measured in dollars and x is measured in units of a bounded. Find the equilibrium quantity and price.

A ladder 25 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second.

Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.

The demand function for Sportsman $5 \times 7$ tents is given by $p=f(x)=-0.1 x^{2}-x+40$ where p is measured in dollars and x is measured in units of a thousand.

What is the rate of change of the unit price if the quantity demanded is 5000?

A spherical balloon is inflated with helium at the rate of 100 pi ft^3/min. How fast is the balloon’s radius increasing at the instant the radius is 5 ft? How fast is the surface area increasing?

The radius of a spherical balloon can be expressed as $r=\sqrt[3]{\frac{3 V}{4 \pi}}$ inches, where r is the radius and V is the volume of the balloon in cubic inches. If air is pumped to inflate the balloon from 500 cubic inches to 800 cubic inches, by how many inches has the radius of the balloon increased?

What was the radius of the balloon originally?

What was the radius after inflating the balloon to 800 cubic inches?

How can you use the two radii to find the amount of increase?

Moving Ladder A ladder $25$ feet long is leaning against the wall of a house (see figure on next page). The base of the ladder is pulled away from the wall at a rate of $2$ feet per second.
Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is $7$ feet from the wall.

The demand function for Sportsman $5 \times 7$ tents is given by $p=f(x)=-0.1 x^{2}-x+40$ where p is measured in dollars and x is measured in units of a thousand.

Find the average rate of change in the unit price of a tent if the quantity demanded is between 5000 and 5050 tents; between 5000 and 5010 tents.

Solve the problems in related rates.

A metal cube dissolves in acid such that an edge of the cube decreases by $0.50$ mm/min. How fast is the volume of the cube changing when the edge is $8.20$ mm?