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

p = −0.2x2 + 80

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 = 0.1x2 + 2.5x + 60

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.)

consumers' surplus $ ?

producers' surplus $ ?

To find the consumers' surplus and the producers' surplus, we first need to find the equilibrium price and quantity. The equilibrium occurs when the quantity demanded equals the quantity supplied. We can find this point by setting the demand and supply equations equal to each other and solving for x:

-0.2x^2 + 80 = 0.1x^2 + 2.5x + 60

Combining like terms:

0.3x^2 + 2.5x - 20 = 0

Now we can solve this quadratic equation using any method, such as factoring or the quadratic formula. In this case, let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For this equation, a = 0.3, b = 2.5, and c = -20. Plugging these values into the formula:

x = (-2.5 ± √(2.5^2 - 4*0.3*(-20))) / (2*0.3)

x = (-2.5 ± √(6.25 + 24)) / 0.6

x = (-2.5 ± √30.25) / 0.6

x = (-2.5 ± 5.5) / 0.6

Now we have two possible values for x:

x1 = (-2.5 + 5.5) / 0.6 = 3.33 (rounded to 2 decimal places)

x2 = (-2.5 - 5.5) / 0.6 = -13.33 (rounded to 2 decimal places)

Since x represents the quantity in hundreds, we discard the negative value. Therefore, the equilibrium quantity is approximately 3.33 * 100 = 333 units.

To find the equilibrium price, we can substitute this value of x into either the demand or supply equation. Let's use the supply equation:

p = 0.1x^2 + 2.5x + 60

p = 0.1(333^2) + 2.5(333) + 60

p = 0.1(110889) + 2.5(333) + 60

p = 11088.9 + 832.5 + 60

p = 11981.4 (rounded to the nearest dollar)

Therefore, the equilibrium price is approximately $11,981.

To find the consumers' surplus, we need to calculate the area of the triangle formed by the demand curve and the price axis up to the equilibrium quantity. The formula for the area of a triangle is:

Consumers' surplus = 0.5 * base * height

The base is the equilibrium quantity (333) and the height is the difference between the equilibrium price (11981) and the price axis (0). Plugging these values into the formula:

Consumers' surplus = 0.5 * 333 * 11981

Consumers' surplus = 1,985,876.5 (rounded to the nearest dollar)

Therefore, the consumers' surplus is approximately $1,985,877.

To find the producers' surplus, we need to calculate the area of the triangle formed by the supply curve and the price axis up to the equilibrium quantity. Using the same formula:

Producers' surplus = 0.5 * base * height

The base is the equilibrium quantity (333) and the height is the difference between the equilibrium price (11981) and the price axis (0). Plugging these values into the formula:

Producers' surplus = 0.5 * 333 * 11981

Producers' surplus = 1,985,876.5 (rounded to the nearest dollar)

Therefore, the producers' surplus is approximately $1,985,877.

Therefore, the consumers' surplus is $1,985,877 and the producers' surplus is $1,985,877.