The quantity demanded x (in units of a hundred) of the Mikado miniature cameras per week is related to the unit price p (in dollars) by

p = −0.2x^2 + 220
and the quantity x (in units of a hundred) that the supplier is willing to make available in the market is related to the unit price p (in dollars) by
p = 0.1x^2 + 8x + 110.
If the market price is set at the equilibrium price, find the consumers' surplus and the producers' surplus. (Round your answers to the nearest dollar.)
consumer's surplus = ?
producer's surplus = ?

To find the consumer's surplus, we first need to solve for the equilibrium price and quantity, where the quantity demanded equals the quantity supplied.

Step 1: Set the two equations equal to each other and solve for x:
-0.2x^2 + 220 = 0.1x^2 + 8x + 110

Simplifying the equation:
-0.2x^2 - 0.1x^2 - 8x + 220 - 110 = 0

Combining like terms:
-0.3x^2 - 8x + 110 = 0

Step 2: Solve the quadratic equation for x by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)

In our equation, a = -0.3, b = -8, and c = 110.

Substituting the values into the formula, we get:
x = (-(-8) ± √((-8)^2 - 4(-0.3)(110)))/(2(-0.3))
x = (8 ± √(64 + 132))/(0.6)
x = (8 ± √(196))/(0.6)
x = (8 ± 14)/(0.6)

Simplifying further, we have two possible solutions:
x1 = (8 + 14)/(0.6) ≈ 37.33
x2 = (8 - 14)/(0.6) ≈ -10.67

Since x represents the quantity demanded and supplied in units of a hundred, the negative value is not meaningful in this context. So we can discard x2 ≈ -10.67.

Therefore, the equilibrium quantity is approximately x ≈ 37.33 (units of a hundred).

Step 3: Substitute the equilibrium quantity into either equation to find the equilibrium price.

Using the first equation, p = -0.2(37.33)^2 + 220:
p ≈ $131.46

So the equilibrium price is approximately $131.46.

Step 4: Calculate the consumer's surplus.

Consumer's surplus is the difference between what consumers are willing to pay (based on the demand function) and what they actually pay at the equilibrium price.

To calculate the consumer's surplus, we need to integrate the demand function from 0 to the equilibrium quantity (37.33) to find the area under the demand curve.

The demand function is given by p = -0.2x^2 + 220.

Integrating the demand function:
∫(-0.2x^2 + 220) dx = [-0.2 * (1/3)x^3 + 220x] evaluated from 0 to 37.33

Consumer's surplus = [-0.2 * (1/3)(37.33)^3 + 220(37.33)] - [-0.2 * (1/3)(0)^3 + 220(0)]
≈ $672.59

Therefore, the consumer's surplus is approximately $672.59.

Step 5: Calculate the producer's surplus.

Producer's surplus is the difference between what producers receive for their goods at the equilibrium price and the minimum price they are willing to accept.

To calculate the producer's surplus, we need to integrate the supply function from 0 to the equilibrium quantity (37.33) to find the area above the supply curve.

The supply function is given by p = 0.1x^2 + 8x + 110.

Integrating the supply function:
∫(0.1x^2 + 8x + 110) dx = [0.1 * (1/3)x^3 + 4x^2 + 110x] evaluated from 0 to 37.33

Producer's surplus = [0.1 * (1/3)(37.33)^3 + 4(37.33)^2 + 110(37.33)] - [0.1 * (1/3)(0)^3 + 4(0)^2 + 110(0)]
≈ $7,914.42

Therefore, the producer's surplus is approximately $7,914.42.

To find the consumers' surplus and the producers' surplus, we need to understand what these terms mean in the given context.

Consumers' Surplus:
Consumers' surplus represents the difference between the price consumers are willing to pay for a product and the actual price they pay. It measures the benefit that consumers receive from purchasing a product at a lower price than what they are willing to pay.

Producers' Surplus:
Producers' surplus represents the difference between the price at which producers are willing to supply a product and the actual price they receive. It measures the benefit that producers receive from selling a product at a higher price than what they are willing to sell for.

To find the equilibrium price, we need to set the demand (consumers' perspective) equal to the supply (producers' perspective):

Demand: p = -0.2x^2 + 220
Supply: p = 0.1x^2 + 8x + 110

Setting these equations equal to each other:
-0.2x^2 + 220 = 0.1x^2 + 8x + 110

Simplifying the equation:
-0.2x^2 - 0.1x^2 + 8x = 110 - 220
-0.3x^2 + 8x = -110

Rearranging and setting the equation equal to zero:
-0.3x^2 + 8x + 110 = 0

To solve the quadratic equation, we can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For the equation -0.3x^2 + 8x + 110 = 0, the coefficients are:
a = -0.3
b = 8
c = 110

Using the quadratic formula:
x = (-8 ± √(8^2 - 4(-0.3)(110))) / (2(-0.3))

Simplifying:
x = (-8 ± √(64 + 132)) / (-0.6)
x = (-8 ± √196) / (-0.6)
x = (-8 ± 14) / (-0.6)

We have two possible solutions:
1) x = (6 / 0.6) = 10
2) x = (-22 / 0.6) = -36.67

Since the number of units cannot be negative, we discard the second solution.

Now that we have the value of x, we can substitute it into either the demand or supply equation to find the equilibrium price (p):

Using the demand equation: p = -0.2x^2 + 220
p = -0.2(10)^2 + 220
p = -20 + 220
p = 200

So, the equilibrium price is $200.

To find the consumers' surplus, we need to calculate the area between the demand curve and the price line. Since the demand curve is a downward-sloping parabola, the consumers' surplus can be found by integrating the area under the curve from 0 to x:

Consumers' Surplus:
∫ (demand equation) dx from 0 to x

∫ (-0.2x^2 + 220) dx from 0 to 10

Integrating:
[-(0.2/3)x^3 + 220x] from 0 to 10

Substituting the values:
[-(0.2/3)(10)^3 + 220(10)] - [-(0.2/3)(0)^3 + 220(0)]

Simplifying:
[-(0.2/3)(1000) + 2200] - [-(0.2/3)(0) + 0]

[-(20/3)(100) + 2200] - [0 + 0]

[-(2000/3) + 2200] - [0]

[-666.67 + 2200] - [0]

1533.33

The consumers' surplus is approximately $1533.

To find the producers' surplus, we need to calculate the area between the supply curve and the price line. Since the supply curve is an upward-sloping parabola, the producers' surplus can be found by integrating the area above the curve from 0 to x:

Producers' Surplus:
∫ (supply equation) dx from 0 to x

∫ (0.1x^2 + 8x + 110) dx from 0 to 10

Integrating:
[(0.1/3)x^3 + 4x^2 + 110x] from 0 to 10

Substituting the values:
[(0.1/3)(10)^3 + 4(10)^2 + 110(10)] - [(0.1/3)(0)^3 + 4(0)^2 + 110(0)]

Simplifying:
[(0.1/3)(1000) + 4(100) + 1100] - [(0.1/3)(0) + 0 + 0]

[(100/3) + 400 + 1100] - [0 + 0 + 0]

[400/3 + 400 + 1100] - [0]

3066.67

The producers' surplus is approximately $3067.

Therefore,
Consumer's surplus ≈ $1533
Producer's surplus ≈ $3067.