Suppose x varies directly with y and x=24 when y =-96. what is the constant of variation and what is the value of x when y=-164?
2 answers:
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Answer:
Step-by-step explanation:
Given :
Solution :
Part A:
First find the potential roots of p(x) using rational root theorem;
So,
Since constant term = -60
Leading coefficient = 1
Thus the possible roots are
Thus from the given options the correct answers are -10,-5,3,15
Now For Part B we will use synthetic division
Out of the possible roots we will use the root which gives remainder 0 in synthetic division :
Since we can see in the figure With -5 we are getting 0 remainder.
Refer the attached figure
We have completed the table and have obtained the following resulting coefficients: 1 , 1,−12,0. All the coefficients except the last one are the coefficients of the quotient, the last coefficient is the remainder.
Thus the quotient is
And remainder is 0 .
So to get the other two factors of the given polynomial we will solve the quotient by middle term splitting
Thus x-3 and x+4 are the other two factors
So , p(x)=(x+5)(x-3)(x+4)
the "t" variable is independent, namely it can change values "freely", and a(t) is dependent, because it depends on those values on "t" to get its own value.
if we make a table of values for minutes, we know the balloon is inflating at 6000 ft³ each passing minute.
1 minute......................... 6000(1) ft³
2 minutes......................6000(2) ft³
3 minutes......................6000(3) ft³
4 minutes......................6000(4) ft³
t minutes......................6000(t) ft³
before getting in the domain and range, let's find out how much air after 6 minutes.
a(6) = 6000(6)
a(6) = 36000 ft³.
now, what domain, namely values for "t" make sense?
well, we know the balloon can only hold up to 90000 ft³, and on every passing minute is filling up by 6000 ft³, after how many minutes will it be filled up?
90000/6000 = 15, after 15 minutes.
so at 0 minutes, t = 0, the balloon is empty, and 15 minutes later, t = 15, the balloon is filled up, [ 0 , 15 ].
what values for the range, namely a(t), makes sense?
well, we know when the balloon is empty is holding 0 ft³, and when is full it has 90000 ft³, [0 , 90000].
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.