QUERY 15

course MTH 271

10/23, 1700

assignment #015015. `query 15

Applied Calculus I

10-23-2009

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20:47:46

2.3.32 P=22t^2+52t+10000, t from 1970; find P at t=0,10,20,25 and explain; find dP/dt; evaluate at given t and explain your results.

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RESPONSE -->

P=22t^2 + 52t + 10000

P(0)=10000

P(10)=12720

P(20)=19840

P(25)=25050

dP/dt=44t + 52

dP/dt(0)=52

dP/dt(10)=492

dP/dt(20)=932

dP/dt(25)=1152

confidence rating:

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20:48:18

dP/dt=44t + 52 (power function rule on each nonconstant term)

When t = 0, 10, 20 and 25 you would have P = 10,000, 12,700, 20,000, 25,000 approx.

At these values of t we have dP / dt = 52, 492, 932 and 1152 (these are my mental calculations--check them).

dP / dt is the rate of change of the population with respect to time t **

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RESPONSE -->

self critique rating: 3

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20:58:57

2.3.48 demand fn p = 50/`sqrt(x), cost .5x+500. Find marginal profit for x=900,1600,2500,3600

Explain how you found the marginal profit, and give your results.

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RESPONSE -->

P=50 / x^1/2

C=0.5x + 500

R=xp

R=x(50 / x^1/2)

R=50 x^1/2

P=R - C

P=(50x^1/2) - (.5x + 500)

dP/dx=25x^-1/2 - 0.5

dP/dx(900)=$.33/unit

dP/dx(1600)=$.13/unit

dP/dx(2500)=$ 0/unit

dP/dx(3600)=$-.08/unit

P(2500)= 50 / (2500)^1/2=$1

confidence rating:

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20:59:15

x represents the number of items sold. If x items are sold at price p = 50 / `sqrt(x), then revenue is price of item * number sold = 50 / `sqrt(x) * x = 50 `sqrt(x).

The profit is revenue - cost = 50 `sqrt(x) - .5 x - 500.

The marginal profit is the derivative of the profit function, which is

(50 `sqrt(x) - .5 x - 500 ) ' = 25 / `sqrt(x) - .5.

Evaluating the marginal profit at x = 900, 1600, 2500 and 3600 we get values

.33..., .125, 0 and -.0833... .

This shows us that the marginal profit, which is the limiting value of the increase in profit per additional item manufactured, is positive until x = 2500. This means that it is to the advantage of the producer to produce new items when x = 900 and when x = 1600, but that the advantage disappears as soon as x reaches 2500.

So 2500 is the best selling price.

When x = 3600 production of additional items reduces profits. **

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RESPONSE -->

self critique rating: 3

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&#Very good responses. Let me know if you have questions. &#

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