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Drive Sounds determines that in order t0 sell units of a new car audio receiver, the price per unit, in dollars_ must be p(x) 1750 It also determines that the total...

Question

Drive Sounds determines that in order t0 sell units of a new car audio receiver, the price per unit, in dollars_ must be p(x) 1750 It also determines that the total cost of producing units is given by C(x) 2250 I5x Find the total revenue R(x) Find the total profit, P() How many units must be made and sold in order tO maximize profit? What is the maximum profit? What price per unit yields this maximum profit?

Drive Sounds determines that in order t0 sell units of a new car audio receiver, the price per unit, in dollars_ must be p(x) 1750 It also determines that the total cost of producing units is given by C(x) 2250 I5x Find the total revenue R(x) Find the total profit, P() How many units must be made and sold in order tO maximize profit? What is the maximum profit? What price per unit yields this maximum profit?



Answers

MARKET RESEARCH The demand $x$ and the price $p$ (in dollars) for portable iPod speakers at a national electronics store are related by $$ x=f(p)=5,000-100 p \quad 0 \leq p \leq 50 $$ The revenue (in dollars) from the sale of $x$ units and the cost (in dollars) of producing $x$ units are given, respectively, by $$ R(x)=50 x-\frac{1}{100} x^{2} \quad \text { and } \quad C(x)=20 x+40,000 $$ Express the profit as a function of the price $p$ and find the price that produces the largest profit.

The solution to the question is here the function is given as -0.04 x Plus 800. Where this x goes from zero to 20,000. So now moving towards party where we have to find the revenue function. So by the definition, the revenue function is given by R. S equals two X and two P. Therefore our affects will be equal to X and 2 -0.04 X. Last 800 which will be equal to minus 0104 X square plus 800 X. And so this will be the answer for the part A. Now moving towards part B where we have to find the marginal revenue that is R dash at so day by day X of Rx that is minus 0.4 X square plus 800 tax. And solving this differential, you will get your answer as minus 0.8 X plus 800. And so this will be the answer for the part to be Now in part C, we have to find our dash of 5000 that is minus 0.8 into 5000. Less 800 will be equal to 400. So that's the Children knew to be realized from the sale of 5001 speaker system is approximately $400. Thank you.

Were given that the revenue are received for selling ex stereos is given by the formula and were asked how many stereos must be sold to obtain the maximum revenue. We know that the maximum revenue because this is a quadratic that opens down has to correct overtax. And I confined the X coordinate of that Vertex by using peat formula the opposite of be over to a So, in this case, to find the X coordinate of the Vertex, I have negative 80 divided by two times negative 1/5. Now I can use, um instead of using negative 1/5 I could use negative zero point to. That makes it easier to do on the calculator, and that should give me the answer of 200. So there are 200 stereos, then were asked, What is the revenue If I substitute acts into my original as 200 again using my calculator to do that computation, I should get that. The maximum revenue is $7000

Were asked to use the methods of this section to answer a question about cost, revenue and profit. We're told that the revenue and cost equations for a product are okay. Are the revenue equals x Times 75 minus 0.5 x and see The cost equation is 30 x plus 250,000 where both are and see are measured in dollars and X represents the number of units sold. Okay were asked how many units must be sold to obtain a profit of at least $750,000 and then were asked, what is the price per unit now? First to find the number of units sold to obtain a profit of at least $750,000 we first want to find an equation for the profit. So I called it the Prophet P is three revenue ar minus the cost. Putting these equations together. This is X Times 75 minus 0.5 X minus 30 X plus 250,000. Yeah. Okay, now we can simplify this equation. This is the same as so we have 75 x minus 30 x We have negative point 000 five X squared plus 45 x minus 250,000. So now we have this equation for the profit. Since our prophet must be at least 750,000, we want to solve the nonlinear inequality. He is greater than or equal to 750,000 Solve this inequality well, notice that this is the same as the inequality. Negative 0.5 X squared plus 45 x minus 250,000 is greater than or equal to 750,000. Now we have a polynomial inequality. To solve this inequality, we want to isolate a polynomial on one side and zero on the other. Then you want to find the zeros of this polynomial. Then we want to test values lying in the intervals between the zeros. And finally, we want to determine whether or not the intervals lie in the solution set so we can rewrite this as negative point 0005 X squared plus 45 x and then we have minus boy of 250,000, minus 750,000 which is minus one million, is greater than or equal to zero. Okay, now this is a quadratic polynomial to find the zeros and use the quadratic inequality. Sorry. Quadratic formula. This is the zeros air going to be negative. 45 plus or minus. The square root of 45 squared minus four times negative 0.0 Five times one million all over two times negative. 20.5 Now simplify this. The zeros are approximately 45.97 and 174 0.3 to two decimal places. Notice that our inequality here is not strict. Therefore, the zeros are also solutions to the inequality. Yeah. Now, with these two zeros, these are the only two zeros of our quadratic polynomial. So we can rewrite it as negative 20.5 times. And then we have the two factors X minus approximately 45.97 and x minus 174.3 And the inequality is still greater than or equal to zero. Of course, we can simplify by dividing both sides by negative 0.5 Because this is less than zero. We have to switch the sign of the inequality. So this becomes x minus 45.97 times X minus 174.3 is now less than or equal to zero. So we have this new form of our inequality Now, going back to our zeros. These two zeros divide the real number line into three parts. So the first part, these air all the numbers less than the smaller zero. 45 0.97 As a test value, consider X equals 45. We plug this in. We have see 45 minus 45.97 times 45 minus 174 0.3 Now it's clear that this first term is going to be less than zero and the second term is also going to be less than zero. Therefore, it follows that the product of these two terms is going to be greater than zero. Therefore, the test value is not a solution. Polynomial is only change signs at their zeros and therefore it follows that none of the points in this interval our solutions second reason to consider are all points between 0 45.97 and the other zero. 174.3 To test value, consider 46. Plug this in. We have 46 minus 45.97 and then 46 minus 174.3 Now it's clear that this first factor is going to be greater than zero. The second factor will be less than zero. So if a product of a positive and a negative and therefore the product is going to be less than zero as well. Therefore, the test value is a solution to our inequality again because this is a polynomial inequality. It follows that all the values in this interval our solutions. Finally, we consider the values greater than 174.3 As a test value. Consider X equals 175. Well, we have 40 Sorry. 175 minus 45.97 times 1 75 minus 1 74 0.3 Now, clearly the first factor is going to be created than zero. And so the second factor and therefore the product is also going to be greater than zero. Therefore, the test values not a solution and by the previous discussion none of the points in the interval are either. So putting this together we hit the solution. Sets of this inequality is going to be well. We have left bracket 45 point 97 since this is a solution and then 1 74.3 right bracket. Since this is also a solution, however, let's recall what X represents X is the number of units sold. Therefore, X must be a natural number. So we have to sell a positive number of units or, of course, zero units. This means that numbers like 45.97 which are fractional, are not solutions. So to remedy this, we simply change our answer so that the end points are both integers which lie inside the interval. So we have that 45.97 is of course closest to the image of 46 but still less than it. And 174 0.3 is closest to, but still greater than 174. And therefore it follows that the number of units must be sold to obtain a profit of at least $750,000 is going to be a number of units between 46 and 174. That's our answer in set notation. Now we're asked to find the price per unit No.

They can find the profit by using a profit equation of revenue minus costs, P equals r minus C. If we are given that a revenue equation Is equal to x times 75 0 05 X. And the cost equation is 30 X Plus 250 On 250,000. These are both going to be expressed in dollars and X represents the number of units sold. And we can combine these two for profit for P is equal to X times 75 minus zero point 0005 x minus 30 x plus $250. And we want to find how many units to obtain a profit of at least 750,000. So we can set this up as 75 X. My anus 0.0005 x squared minus 30 X minus 250,000 Is greater than or equal to 750,000. And here we can simplify this to -0.0005 x squared Plus 45 x -250000. So oneness Combine the 250 and subtract the 750,000 as well. 21 million. Greater than equal to zero. And here we can grab this or use your graph on lesbos dot com. And here with drafting it we can see that the lowest. Mhm The lowest amount is 40,000. You have to sell between 40,000 and 50,000 units. We can find the price per unit. Using the formula P equals are over X. And this is a small p Ankle 7 5 0.0005 x. And For X is equal to 40,000. Yeah, you can substitute that in and have 75 -0.0005 times 40,000 Right equals $55 per unit. And when X is equal to 50,000 75 -0.0005 times 50,000 As equal to $50 per unit, So the price per unit will be between 50 And $55.


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