The Height of a Hill (In Feet) Is Given By h

question 40

Multiple Choice

The height of a hill (in feet) is given by
$$h ( x , y ) = 30 \left( 14 - 5 x ^ { 2 } - 2 y ^ { 2 } + 3 x y + 30 x - 18 y \right)$$
where $x$ is the distance (in miles) east and $y$ is the distance (in miles) north of your cabin. If you are at a point on the hill 1 mile north and 1 mile east of your cabin, what is the rate of change of the height of the hill (a) in a northerly direction and (b) in an easterly direction?

• A) (a) \(570 \mathrm { ft } / \mathrm { mi }\), (b) \(690 \mathrm { ft } / \mathrm { mi }\)
• B) (a) \(- 570 \mathrm { ft } / \mathrm { mi }\), (b) \(690 \mathrm { ft } / \mathrm { mi }\)
• C) (a) \(690 \mathrm { ft } / \mathrm { mi }\), (b) \(570 \mathrm { ft } / \mathrm { mi }\)
• D) (a) \(690 \mathrm { ft } / \mathrm { mi }\), (b) \(- 570 \mathrm { ft } / \mathrm { mi }\)

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Definitions:

Desert Island

A secluded and uninhabited island, often used in scenarios and thought experiments in economics and ethics to explore human behavior in isolation.

Competitive Equilibrium

A market situation where supply equates demand, leaving no incentive for price changes, assuming all agents are price-takers.

Perfect Substitutes

Items that can replace each other without affecting the consumer's satisfaction negatively.

Perfect Complements

Goods that are consumed together in fixed proportions because the consumption of one enhances the value or utility of the other.