Consumers' Surplus for Calculus

**AntoninaFinn** · December 8th 2012, 12:18 PM

The demand function for a certain make of replacement cartridges for a water purifier is given by:
p=-0.01x^2-0.1x+6
where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.
Determine the consumers' surplus if the market price is set at $4/cartridge.

I know the answer is $11,667. How did they get this answer? Particularly, how would I find x for the bounds when I integrate? Please show steps in a thorough, understandable manner.
(The problem involves integration.)
Thank you.

(If anyone asks for a supply function or any second equation -- there is none. This is how the question has been exactly written in the book.)

**richard1234** · December 8th 2012, 01:15 PM

Problem makes no sense. Why is the problem asking for a surplus (units: cartridges) when the answer is a dollar amount?

**AntoninaFinn** · December 8th 2012, 02:14 PM

Why would my book make me solve a problem that doesn't make sense? (That's also a question I'm asking myself since you say the problem makes no sense.) But that's exactly how the problem is written out in my book. I double-checked the answer and it is $11,667.

My book shows me how to solve problems like these (producers' and consumers' surplus) when they give me two equations (which I set them to each other, then set the resulting equation to 0 to find x - quantity demanded), but not what to do when I have only one equation.
So, I took this equation, set it to 0 to find x, but gave me a weird number, which I used anyway. It didn't work. Or maybe it was the right number, but I did something else wrong.

Someone has to know what do. There has to be someone who can help me.

**richard1234** · December 8th 2012, 02:35 PM

Originally Posted by AntoninaFinn

So, I took this equation, set it to 0 to find x, but gave me a weird number, which I used anyway. It didn't work. Or maybe it was the right number, but I did something else wrong.

You can't just set the RHS of that equation to zero. That would be like, solving for the demand when the price is $0. If we let p = 4 and solve, we obtain x = 10 (10,000 cartridges).

**AntoninaFinn** · December 8th 2012, 03:34 PM

But that's not the answer in the book. *Smiles* So, are you saying that the book's answer is wrong?

And so, if I set 4 to the equation, it would be:
4= -0.01x^2-0.1x+6
Subtract 4 to get to the other side, which would make: -.0.01x^2-0.1x+2
Is that what you mean?

Thank you.

**richard1234** · December 8th 2012, 04:22 PM

Sorry, I might've mis-understood! I didn't know the difference between consumers' surplus and a surplus (they're two completely different things).

According to Wikipedia, a consumers' surplus is "the monetary gain obtained by consumers because they are able to purchase a product for a price that is less than that they would be willing to pay." Using this definition, we find the highest price that consumers are willing to pay by maximizing $p$ . To do this, we graph p and see that the highest p that produces a non-negative demand is p = 6.

The consumers' surplus is equal to

$S = \int_{4}^{6} x(p)\, dp$ where x(p) is the demand as a function of price. To find x(p) we must solve for x in the equation:

$p = -0.01x^2 - 0.1x + 6 \Rightarrow 0.01x + 0.1x + (p-6) = 0$

$x = \frac{-0.1 + \sqrt{0.01 - 4(0.01)(p-6)}}{0.02} = -5 + 5\sqrt{25 - 4p}$ (taking positive root).

However x is in thousands so we should multiply this expression by 1000 to get $1000(-5 + 5\sqrt{25 - 4p})$ . Integrating from p = 4 to p = 6,

$S = \int_{4}^{6} 1000(-5 + 5\sqrt{25-4p}) \, dp = \frac{35000}{3} = 11667$ .

Aha! Previously I didn't know the difference between "consumers' surplus" and "surplus." Sorry for the prior confusion.

**skeeter** · December 8th 2012, 05:58 PM

Aha! Previously I didn't know the difference between "consumers' surplus" and "surplus." Sorry for the prior confusion.

No need for an apology ... it is incumbent on the original poster to clearly interpret all contextual terms used in posting a problem.

**AntoninaFinn** · December 9th 2012, 12:00 PM

I'm sorry, but I'm still confused. I've been going over the quadratic equation many times over and trying to figure out how you received -5+5(sq/25-4p)

I've never solved a quadratic equation with an actual letter in it, so I was confused as to how to go about it. I thought to substitute 1 for a p so it would be (1-6), which within the square root of 0.1^2-4(0.01)(p-6) would equal square root 0.21
Many thanks if you would explain this part to me.

And don't worry about confusing the "surplus" aspect, we all mess up sometimes.
Good day.

**skeeter** · December 9th 2012, 12:17 PM

$p=-0.01x^2-0.1x+6$

$0.01x^2 + 0.1x + p - 6 = 0$

$a = 0.01$

$b = 0.1$

$c = p - 6$

$x = \frac{-0.1 + \sqrt{(0.1)^2 - 4(0.01)(p-6)}}{2(0.01)}$

$x = \frac{-0.1 + \sqrt{0.01 - 4(0.01)(p-6)}}{0.02}$

$x = \frac{-0.1 + \sqrt{0.01[1 - 4(p-6)]}}{0.02}$

$x = \frac{-0.1 + 0.1\sqrt{25 - 4p}}{0.02}$

$x = \frac{-0.1[1 - \sqrt{25 - 4p}]}{0.02}$

$x = -5 \left(1 - \sqrt{25-4p} \right)$

$x = 5 \left(\sqrt{25-4p} - 1 \right)$

**richard1234** · December 9th 2012, 12:40 PM

Originally Posted by AntoninaFinn

I'm sorry, but I'm still confused. I've been going over the quadratic equation many times over and trying to figure out how you received -5+5(sq/25-4p)

Ah, good algebra skills are essential for a calculus course. Just use the quadratic formula and substitute a = 0.01, b = 0.1, c = p-6 as skeeter did.

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