a company\'s market research department recommends the manufacture and marketing

Question

a company's market research department recommends the manufacture and marketing of a new headphone set for MP3 players. after suitable test marketing, the research department presents the following price demand equation: p=10-0.0011x where x is the number of headphones that retailers are likely to buy at $p per set. the financial department provides the cost function c(x)=7,000 + 2x where 7,000 is the estimate of fixed costs (tooling and overhead) and $2 is the estimate of variable costs per headphone set (materials,labor, marketing, transporation,storage, etc).
A- given the price demand equation for what values of x is it appropriate?
B- fina and interpret the marginal cost function c' (x)
C- find the revenue function as a function of x. for what values of x is it appropriate?
D- find the marginal revenue at x=2,000,5,000, and 7,000. interpret these results
E-graph the cost function and the revenue function in the same coordinate system. find the intersection points of these two graphs and interpret the results.
F-find the profit function and sketch the graph of the function. for what values of x is it appropriate?
G- find the marginal profit at x=1,000,4,000, and 6,000. interpret these results.

Explanation / Answer

(A)

p = 10 - 0.0011x

Since price can never be negative, the lowest price can be 0. When p = 0,

0 = 10 - 0.0011x

0.0011x = 10

x = 10 / 0.0011 = 9,091

Since x (quantity) is non-negative, lowest value for x is 0.

Acceptable range of x is:

0 < x < 9,091

(B)

Marginal cost, MC = dC / dx = 2

This means that as output increases by 1 unit, Cost increases by $2.

(C)

Revenue, TR = p.x = 10x - 0.0011x2

The lowest value of revenue is 0. When TR = 0,

10x - 0.0011x2 = 0

x. (10 - 0.0011x) = 0

Either x = 0, or x = 9,091

(D)

Marginal revenue, MR = dTR / dx = 10 - 0.0022x

When x = 2,000: MR = 10 - (0.0022 x 2,000) = 10 - 4.4 = 5.6

This means that, as output increases by 1 unit, total revenue increases by $5.6.

When x = 5,000: MR = 10 - (0.0022 x 5,000) = 10 - 11 = - 1

This means that, as output increases by 1 unit, total revenue decreases by $1.

When x = 7,000: MR = 10 - (0.0022 x 7,000) = 10 - 15.4 = - 5.4

This means that, as output increases by 1 unit, total revenue decreases by $5.4.

Note: First 4 sub-parts are answered.

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