Given the following Demand Curve: P = 24 - 8Q
1 answer:
Answer:
Maximum value of total revenue = 18
Step-by-step explanation:
- Express this Demand Curve in terms of Q.
<em>P = 24 - 8Q (it is the same equation)</em>
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- Find P, if Q=2
<em>P= 24-8(2)</em>
<em>P=24-16</em>
<em>P=8</em>
- Find Q if P = 8.
<em>P = 24 - 8Q</em>
<em>8= 24 - 8Q</em>
<em>8Q= 24-8</em>
<em>8Q=16</em>
<em>Q=16/8</em>
<em>Q=2</em>
- Total revenue in terms of Q
<em>Total revenue is P times Q, that is</em>
<em>P*Q=TR=(24-8Q)*Q</em>
<em>TR=24Q-8Q^2</em>
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- Marginal Revenue
<em>It is the first derivative of TR</em>
<em>TR'(Q)= 24-16Q</em>
- Find the values of P and Q that will maximize total revenue.
<em>To find them first TR'(Q)=0, that is</em>
<em>0=24-16Q</em>
<em>16Q=24</em>
<em>Q=24/16</em>
<em>Q=3/2</em>
<em>Q=1.5</em>
<em>and we plug in 1.5 in P=24-8Q, which is,</em>
<em>P=24-8Q</em>
<em>P=24-8(1,5)</em>
<em>P=24-12</em>
<em>P=12</em>
- Calculate this maximum value of total revenue
<em>P=12 Q=1.5</em>
<em>P*Q=Total Revenue</em>
<em>12*1.5=Total Revenue</em>
<em>18=Total Revenue</em>
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Do the right first. It's the one that you have to pay the most attention to because it is any intuative. y = 1/(x - 7) That looks like it should shift to the left. But it does not. Take a simple example
if y = 1/x is used first then if x = 1 then y = 1/1 = 1
if y = 1/(x - 7) is considered then if x = 1 then y = 1/(1 - 7) = -1/6
The question is where does 1/x = -1/6 the answer is when x = -6
So the same answer is obtained with a shift to the right. Try this a couple of times to convince yourself of it.
To shift the graph up, all you need do is add 12.
y = 1/(x - 7) + 12 is your answer. I'll put a graph up to show you how it works.
Notice the blue function. It is shifted up 12 (which you can see from the portion that starts from the left and eventually heads down. And then it is also going right (same part of the blue graph.
The red graph is y = 1/x
if y = 1/x is used first then if x = 1 then y = 1/1 = 1
if y = 1/(x - 7) is considered then if x = 1 then y = 1/(1 - 7) = -1/6
The question is where does 1/x = -1/6 the answer is when x = -6
So the same answer is obtained with a shift to the right. Try this a couple of times to convince yourself of it.
To shift the graph up, all you need do is add 12.
y = 1/(x - 7) + 12 is your answer. I'll put a graph up to show you how it works.
Notice the blue function. It is shifted up 12 (which you can see from the portion that starts from the left and eventually heads down. And then it is also going right (same part of the blue graph.
The red graph is y = 1/x