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Question 1181204: I am trying to solve this problem and unfortunately, I am self-teaching myself so I don't have a lot of direction or advice on how to figure this out. Any tips would be great, thank you!!
[PROBLEM]
Extreme Protection, Inc. manufactures helmets for skiing and snowboarding. The fixed costs for one model of helmet are $7200 per month. Materials and labor for each helmet of this model are $50, and the company sells this helmet to dealers for $85 each. (Let x represent the number of helmets sold. Let C, R, and P be measured in dollars.)
(a) For this helmet, write the function for monthly total costs C(x).
C(x) =
(b) Write the function for total revenue R(x).
R(x) =
(c) Write the function for profit P(x).
P(x) =
(d) Find C(200).
C(200) =
Interpret C(200).
For every additional helmet produced the cost increases by this much.
This is the cost (in dollars) of producing 200 helmets.
When this many helmets are produced the cost is $200.
For each $1 increase in cost this many more helmets can be produced.
Correct: Your answer is correct.
Find R(200).
R(200) =
Interpret R(200).
For every additional helmet produced the revenue generated increases by this much.
This is the revenue (in dollars) generated from the sale of 200 helmets.
For each $1 increase in revenue this many more helmets can be produced.
When this many helmets are produced the revenue generated is $200.
Incorrect: Your answer is incorrect.
Find P(200).
P(200) =
Interpret P(200).
This is the profit (in dollars) when 200 helmets are sold, but since it is negative it means that the company loses money when 200 helmets are sold.
For each additional helmet sold the profit (in dollars) increases by this much, but since it is negative it means that the company needs to decrease the number of helmets sold in order to make a profit.
For each additional helmet sold the profit (in dollars) increases by this much, but since it is positive it means that the company is producing too many helmets.
This is the profit (in dollars) when 200 helmets are sold, and since it is positive it means that the company makes money when 200 helmets are sold.
(e) Find C(300).
C(300) =
Interpret C(300).
When this many helmets are produced the cost is $300.
For each $1 increase in cost this many more helmets can be produced.
This is the cost (in dollars) of producing 300 helmets.
For every additional helmet produced the cost increases by this much.
Find R(300).
R(300) =
Interpret R(300).
This is the revenue (in dollars) generated from the sale of 300 helmets.
When this many helmets are produced the revenue generated is $300.
For every additional helmet produced the revenue generated increases by this much.
For each $1 increase in revenue this many more helmets can be produced.
Find P(300).
P(300) =
Interpret P(300).
This is the profit (in dollars) when 300 helmets are sold, but since it is negative it means that the company loses money when 300 helmets are sold.
For each additional helmet sold the profit (in dollars) increases by this much, but since it is positive it means that the company is producing too many helmets.
This is the profit (in dollars) when 300 helmets are sold, and since it is positive it means that the company makes money when 300 helmets are sold.
For each additional helmet sold the profit (in dollars) increases by this much, but since it is negative it means that the company needs to decrease the number of helmets sold in order to make a profit.
(f) Find the marginal profit
MP.
MP =
Write a sentence that explains its meaning.
When costs are decreased by this much the profit is increased by $1.
When revenue is increased by this much the profit is increased by $1.
Each additional helmet sold increases the profit by this many dollars.
For each $1 increase in profit this many more helmets can be produced.
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
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.
Answer by ikleyn(52772)  (Show Source):
You can put this solution on YOUR website! .
What you call "self-teaching", is not self-teaching, at all.
I do not see any documented signs of self-teaching in your post.
What I see INSTEAD in your post, is your strong wish to do nothing of your job and to re-load it entirely
to the shoulders of tutors, using their work for free.
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