Question 1181204
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Part (a)


x = number of helmets made and sold
C(x) = 50x+7200


This is because the 50x is from the material/labor aspect of the cost and then we have an additional $7200 added on.

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Part (b)


R(x) = 85x
We simply multiply the cost per helmet with the number of helmets sold


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Part (c)


Profit = Revenue - Costs
P(x) = R(x) - C(x)
P(x) = (85x) - (50x+7200)
P(x) = 85x - 50x-7200
P(x) = 35x-7200


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Part (d)


Plug x = 200 into the C(x) function we found back in part (a)


C(x) = 50x+7200
C(200) = 50(200)+7200
C(200) = 10000+7200
C(200) = 17200
It costs $17,200 to make x = 200 helmets


Repeat for the revenue function
R(x) = 85x
R(200) = 85*200
R(200) = 17,000
If you sold x = 200 helmets, then you earn $17,000 in revenue


Now compute the profit
P(x) = 35x-7200
P(200) = 35*200-7200
P(200) = -200
The negative profit means you lost money if you sold x = 200 helmets. This is because the costs (17200) exceeds the revenue (17000). The business is in the red.


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Part (e)


This is the same as part (d). There's not much to say about this other than the values are different of course.


You should get the following:
C(300) = 22,200
R(300) = 25,500
P(300) = 3300
The interpretations of each are the same as before, again with different values. 
The positive profit means the company is in the black, and is making money at this point.

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Part (f)


The marginal profit is the additional value of profit made when you sell one extra unit. In this case, when one additional helmet is sold, the marginal profit is $35


This is directly from the slope of the profit function (see part (c) above)
P(x) = 35x-7200
The slope is the rate of change. It tells us how much the output P(x) is changing when x increases by 1.


Think of it like this:
slope = rise/run
slope = (change in y)/(change in x)
slope = (change in P(x))/(change in x)
So this is another way to see that bumping x up by 1 will increase the profit by $35. Hence the marginal profit is $35. 


Or simply: the profit per helmet is $35


Note: to find the breakeven point, you'll solve P(x) = 0
So P(x) = 35x-7200 = 0 leads to x = 7200/35 = 205.71; telling the managers that they need to sell 206 or more helmets to have positive profit. This explains why x = 200 helmets made a negative profit. 

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