The demand function for a certain make of replacement cartridges for a water purifier is given by the following equation where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.
p = -0.01x^2 - 0.1 x + 47
Determine the consumers' surplus if the market price is set at $5/cartridge. (Round your answer to two decimal places.)
5 = -0.01x^2 - 0.1 x + 47
0.01x^2 +.1x -47+5 = 0
x^2 + 10x -4200=0
(x-60)(x+ 70)=0
x = 60
x = -70
ʃ (−0.01x^2 − 0.1x + 47)dx on [0,60] - 5*60
(-.01/3x^3/3 -.1x^2/2 + 47x on [0,60] - 300
1920-300 = 1620
Well, let's see how many cartridges they can sell at a price of $5. We'll substitute p = 5 into the demand equation and solve for x:
5 = -0.01x^2 - 0.1x + 47
Now, let's solve this quadratic equation to find the quantity demanded (x).
To determine the consumer's surplus, we need to find the area under the demand curve and above the market price.
Given that the market price is set at $5/cartridge, we can substitute p = 5 into the demand function:
5 = -0.01x^2 - 0.1x + 47
This equation needs to be solved to find the corresponding quantity demanded. Rearranging the equation:
0 = -0.01x^2 - 0.1x + 42
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For this equation, a = -0.01, b = -0.1, and c = 42. Plugging these values into the quadratic formula:
x = (-(-0.1) ± √((-0.1)^2 - 4(-0.01)(42))) / (2(-0.01))
Simplifying,
x = (0.1 ± √(0.01 + 1.68)) / (-0.02)
x = (0.1 ± √1.69) / (-0.02)
x = (0.1 ± 1.3) / (-0.02)
Applying both possibilities:
x = (0.1 + 1.3) / (-0.02) or x = (0.1 - 1.3) / (-0.02)
x = 1.4 / (-0.02) or x = (-1.2) / (-0.02)
x = -70 or x = 60
Since we are looking for a positive quantity demanded, we can discard the negative value. Therefore, x = 60.
To find the consumer's surplus, we need to find the area under the demand curve from x = 0 to x = 60, and above the price of $5.
∫[0,60] (-0.01x^2 - 0.1x + 47 - 5) dx
Integrating this function:
= ∫[0,60] (-0.01x^2 - 0.1x + 42) dx
= [-0.0033x^3 - 0.05x^2 + 42x] [0,60]
= (-0.0033(60)^3 - 0.05(60)^2 + 42(60)) - (0 - 0)
= (71.28 - 180 + 2520) - 0
= 2411.28
Therefore, the consumer's surplus when the market price is set at $5/cartridge is $2411.28.
To determine the consumer surplus, we first need to find the quantity demanded at the market price of $5/cartridge.
Given that the price is $5, we can substitute p = $5 into the demand function equation:
5 = -0.01x^2 - 0.1x + 47
Now we need to solve this quadratic equation for x to find the quantity demanded:
-0.01x^2 - 0.1x + 42 = 0
To solve this quadratic equation, you can use either the quadratic formula or factoring. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a = -0.01, b = -0.1, and c = 42. Plugging in these values:
x = (-(-0.1) ± √((-0.1)^2 - 4(-0.01)(42))) / (2(-0.01))
Simplifying further:
x = (0.1 ± √(0.01 + 1.68)) / (-0.02)
x = (0.1 ± √1.69) / (-0.02)
Now, calculate the two possible values for x:
x1 = (0.1 + √1.69) / (-0.02)
x2 = (0.1 - √1.69) / (-0.02)
Using a calculator, we get:
x1 ≈ -35.53
x2 ≈ 5.53
Since we're dealing with quantities, a negative value doesn't make sense in this context. So we can ignore x1 = -35.53.
Therefore, the quantity demanded at the market price of $5/cartridge is x = 5.53 thousand units.
To calculate the consumer surplus, we need to find the area under the demand curve up to the quantity demanded, which in this case is x = 5.53 thousand units.
Consumer surplus = ∫[0, 5.53] (-0.01t^2 - 0.1t + 47) dt
Evaluating this definite integral from 0 to 5.53:
Consumer surplus = ∫[0, 5.53] (-0.01t^2 - 0.1t + 47) dt
To solve this integral, we can split it into three parts:
Consumer surplus = ∫[0, 5.53] (-0.01t^2) dt + ∫[0, 5.53] (-0.1t) dt + ∫[0, 5.53] (47) dt
Integrating each part separately:
Consumer surplus = [-0.01(1/3)t^3] + [-0.1(1/2)t^2] + [47t] evaluated from 0 to 5.53
Consumer surplus = [-0.01(1/3)(5.53)^3] + [-0.1(1/2)(5.53)^2] + [47(5.53)] - [0]
Now calculate this expression:
Consumer surplus ≈ 120.50
Therefore, the consumer surplus is approximately $120.50.