For a certain company, the cost function for producing x

 items is C(x)=30x+100

 and the revenue function for selling x

 items is R(x)=−0.5(x−90)

2

+4,050

. The maximum capacity of the company is 140

 items.

 

The profit function P(x)

 is the revenue function R(x)

 (how much it takes in)  minus the cost function C(x)

 (how much it spends). In economic models, one typically assumes that a company wants to maximize its profit, or at least make a profit!

 

Answers to some of the questions are given below so that you can check your work.

 

1. Assuming that the company sells all that it produces, what is the profit function? P(x)=  .
2. What is the domain of P(x)? Hint: Does calculating P(x) make sense when x=−10  or x=1,000?
3. The company can choose to produce either 60 or 70 items. What is their profit for each case, and which level of production should they choose?
4. Profit when producing 60 items =    
5. Profit when producing 70 items =   
6. Can you explain, from our model, why the company makes less profit when producing 10 more units?

?1.?

The profit function,

?P(x)=R(x)−C(x)?

?=−0.5(x−90)2+4050−(30x+100)?

?=−0.5(x2−180x+8100)+4050−(30x+100)?

?=−0.5x2+90x−4050+4050−30x−100?

?=−0.5x2+60x−100?

?=−0.5(x2−2×60x)−100?

?=−0.5(x2−2×x×60+602−602)−100?

?=−0.5(x2−2×x×60+602)−0.5×(−602)−100?

?=−0.5(x−60)2−0.5×(−3600)−100?

?=−0.5(x−60)2+1800−100?

?=−0.5(x−60)2+1700??

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?2.?

Since the number of items produced should be non negative, ?x≥0?

Also the maximum capacity of the company is ?140? items, that is ?x≤140?

Therefore for the given situation, the domain of the profit function is

?D={x:0≤x≤140}?

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?4.?

The profit of the company when it produces ?60? items,

?P(60)=−0.5(60−60)2+1700=0+1700=1700??

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?5.?

The profit of the company when it produces ?70? items,

??P(70)=−0.5(70−60)2+1700?

?=−0.5×102+1700?

?=−0.5×100+1700?

?=−50+1700?

?=1650??

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?3.?

Company should choose to produce ?60? items,

because it will be more profitable.

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?6.?

From the model we see that the profit function is

?P(x)=−0.5(x−60)2+1700?

Now for any value of ?x,?

?(x−60)2≥0?−0.5(x−60)2≤0?

??P(x)=−0.5(x−60)2+1700≤1700?

The equality occurs when ?x−60=0?x=60?

That is for number of production ?x=60,?

the company makes maximum profit of ?1700?

But for any other value of ?x? company makes less profit than 1700.

That is why even though company produces more items, it makes less profit.

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