jonathan wants to save up enought money so that he can buy a new sports equipment set that includes a football baseball sccer ba
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Answer:
Part (A)
- 1. Maximum revenue: $450,000
Part (B)
- 2. Maximum protit: $192,500
- 3. Production level: 2,300 television sets
- 4. Price: $185 per television set
Part (C)
- 5. Number of sets: 2,260 television sets.
- 6. Maximum profit: $183,800
- 7. Price: $187 per television set.
Explanation:
<u>0. Write the monthly cost and price-demand equations correctly:</u>
Cost:
Price-demand:
Domain:
<em>1. Part (A) Find the maximum revenue</em>
Revenue = price × quantity
Revenue = R(x)
Simplify
A local maximum (or minimum) is reached when the first derivative, R'(x), equals 0.
Solve for R'(x)=0
Is this a maximum or a minimum? Since the coefficient of the quadratic term of R(x) is negative, it is a parabola that opens downward, meaning that its vertex is a maximum.
Hence, the maximum revenue is obtained when the production level is 3,000 units.
And it is calculated by subsituting x = 3,000 in the equation for R(x):
- R(3,000) = 300(3,000) - (3000)² / 20 = $450,000
Hence, the maximum revenue is $450,000
<em>2. Part (B) Find the maximum profit, the production level that will realize the maximum profit, and the price the company should charge for each television set. </em>
i) Profit(x) = Revenue(x) - Cost(x)
- Profit (x) = R(x) - C(x)
ii) Find the first derivative and equal to 0 (it will be a maximum because the quadratic function is a parabola that opens downward)
- Profit' (x) = -x/10 + 230
- -x/10 + 230 = 0
- -x + 2,300 = 0
- x = 2,300
Thus, the production level that will realize the maximum profit is 2,300 units.
iii) Find the maximum profit.
You must substitute x = 2,300 into the equation for the profit:
- Profit(2,300) = - (2,300)²/20 + 230(2,300) - 72,000 = 192,500
Hence, the maximum profit is $192,500
iv) Find the price the company should charge for each television set:
Use the price-demand equation:
- p(x) = 300 - x/20
- p(2,300) = 300 - 2,300 / 20
- p(2,300) = 185
Therefore, the company should charge a price os $185 for every television set.
<em>3. Part (C) If the government decides to tax the company $4 for each set it produces, how many sets should the company manufacture each month to maximize its profit? What is the maximum profit? What should the company charge for each set?</em>
i) Now you must subtract the $4 tax for each television set, this is 4x from the profit equation.
The new profit equation will be:
- Profit(x) = -x² / 20 + 230x - 4x - 72,000
- Profit(x) = -x² / 20 + 226x - 72,000
ii) Find the first derivative and make it equal to 0:
- Profit'(x) = -x/10 + 226 = 0
- -x/10 + 226 = 0
- -x + 2,260 = 0
- x = 2,260
Then, the new maximum profit is reached when the production level is 2,260 units.
iii) Find the maximum profit by substituting x = 2,260 into the profit equation:
- Profit (2,260) = -(2,260)² / 20 + 226(2,260) - 72,000
- Profit (2,260) = 183,800
Hence, the maximum profit, if the government decides to tax the company $4 for each set it produces would be $183,800
iv) Find the price the company should charge for each set.
Substitute the number of units, 2,260, into the equation for the price:
- p(2,260) = 300 - 2,260/20
- p(2,260) = 187.
That is, the company should charge $187 per television set.
Answer:
See the proof below.
Step-by-step explanation:
What we need to proof is this: "Assuming X a vector space over a scalar field C. Let X= {x1,x2,....,xn} a set of vectors in X, where . If the set X is linearly dependent if and only if at least one of the vectors in X can be written as a linear combination of the other vectors"
Proof
Since we have a if and only if w need to proof the statement on the two possible ways.
If X is linearly dependent, then a vector is a linear combination
We suppose the set is linearly dependent, so then by definition we have scalars
in C such that:
And not all the scalars are equal to 0.
Since at least one constant is non zero we can assume for example that , and we have this:
We can divide by c1 since we assume that and we have this:
And as we can see the vector can be written a a linear combination of the remaining vectors
. We select v1 but we can select any vector and we get the same result.
If a vector is a linear combination, then X is linearly dependent
We assume on this case that X is a linear combination of the remaining vectors, as on the last part we can assume that we select and we have this:
For scalars defined in C. So then we have this:
So then we can conclude that the set X is linearly dependent.
And that complet the proof for this case.