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## Your business sells cupcakes in boxes of 10. The demand equation is x = −5p + 100 In the above formula x is the number of boxes

Question

Your business sells cupcakes in boxes of 10. The demand equation is

x = −5p + 100

In the above formula x is the number of boxes you sell in one month for a unit price (per box) of p dollars. The cost of producing x boxes is

C = $50 + 6x

Set up the profit function P in terms of an arbitrary number of boxes x alone.

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2021-09-10T09:44:41+00:00
2021-09-10T09:44:41+00:00 2 Answers
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## Answers ( )

In Business, the following functions are important.

Revenue function = (price per unit) . (quantity of units) Symbols:

Cost function = (average cost per unit) . (quantity of units) Symbols:

Profit function = revenue − cost

Symbols:

Sometimes in a problem some of these functions are given. Note: Do not confuse p and P .

The price per unit p is also called the demand function p .

Marginal Functions:

The derivative of a function is called marginal function.

The derivative of the revenue function R(x) is called marginal revenue with notation: The derivative of the cost function C(x) is called marginal cost with notation:

The derivative of the profit function P(x) is called marginal profit with notation:

Example 1: Given the price in dollar per unit p = −3×2 + 600x , find: (a) the marginal revenue at x = 300 units. Interpret the result.

R=p.x

C=C.x

P=R−C

R′(x) = dR dx

C′(x) = dC dx

P′(x) = dP dx

revenue function: R(x) = p . x = (−3×2 + 600x) . x = −3×3 + 600×2 marginal revenue: R′(x) = dR = −9×2 + 1200x

dx

marginalrevenueat x=300 =⇒ R′(300)= dR =−9(300)2+1200(300)=−450000

dx x=300

Interpretation: If production increases from 300 to 301 units, the revenue decreases by 450 000 dollars.

(b) the marginal revenue at x = 100 units. Interpret the result. revenue function: R(x) = p . x = (−3×2 + 600x) . x = −3×3 + 600×2

marginal revenue: R′(x) = dR = −9×2 + 1200x dx

marginalrevenueat x=100 =⇒ R′(100)= dR =−9(100)2+1200(100)=30000 dx x=100

Interpretation: If production increases from 100 to 101 units, the revenue increases by 30 000 dollars.

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