Demand for a certain Commodity

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In summary, the demand for a certain commodity is given by the function D(x) = 1000e^-0.03x units per month when the market price is x dollars per unit. The rate of consumer expenditure, E(x) = xD(x), is changing with respect to price x at a rate of E'(x) = 1000e^-0.6x. The consumer expenditure stops increasing and begins to decrease when the derivative E'(x) is equal to 0, and the rate of consumer expenditure begins to increase when E'(x) is positive.
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confusedbycalc
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The demand for a certain commodity is

D(x)  =  1000e−.03x

units per month when the market price is x dollars per unit.

(a) At what rate is the consumer expenditure E(x) = xD(x) changing with respect to price x when the price is equal to $160 dollars?
(b) At what price does consumer expenditure stop increasing and begin to decrease?
(c) At what price does the rate of consumer expenditure begin to increase?

I am not sure but I got -31.27 for a) but i really have no idea how to go about this question.
 
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  • #2
confusedbycalc said:
The demand for a certain commodity is

D(x)  =  1000e−.03x
What is "e" here? (I suspect it is not 2.718...

units per month when the market price is x dollars per unit.

(a) At what rate is the consumer expenditure E(x) = xD(x) changing with respect to price x when the price is equal to $160 dollars?
With D(x)= 1000e- 0.3x, E(x)= 1000ex- 0.3x^2. The rate of change of that is the derivative E'(x)= 1000e- 0.6x.

(b) At what price does consumer expenditure stop increasing and begin to decrease?
As long as E' is positive the consumer expenditure is increasing. It is decreasing when E' is negative. To change from positive to negative, E' has to become 0.

(c) At what price does the rate of consumer expenditure begin to increase?
when is E' positive?

I am not sure but I got -31.27 for a) but i really have no idea how to go about this question.
 

Related to Demand for a certain Commodity

1. What is "demand" for a certain commodity?

Demand refers to the amount of a commodity that consumers are willing and able to purchase at a given price and time. It is one of the key factors that determines the market price of a commodity.

2. What factors affect the demand for a certain commodity?

There are several factors that can affect the demand for a commodity, including consumer preferences, income levels, availability of substitutes, and changes in market trends or demographics. External factors such as economic conditions and government policies can also impact demand.

3. How is the demand for a certain commodity measured?

Demand for a commodity is typically measured by the quantity of the commodity that is purchased at a specific price point. This information is often collected through market research, sales data, and surveys.

4. What is the relationship between price and demand for a certain commodity?

Generally, there is an inverse relationship between price and demand for a commodity. This means that as the price of a commodity increases, the demand for it decreases, and vice versa. However, the exact relationship can vary depending on several factors, such as the type of commodity and consumer behavior.

5. How does the concept of "elasticity" apply to the demand for a certain commodity?

Elasticity is a measure of how responsive the demand for a commodity is to changes in price. If a commodity has high elasticity, a small change in price can result in a significant change in demand. On the other hand, if a commodity has low elasticity, a change in price may have little effect on demand. Elasticity can also be affected by factors such as availability of substitutes and consumer income.

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