Answer: (1 pt.) For the following pair of supply and demand equations, where x r

Question

Answer: (1 pt.) For the following pair of supply and demand equations, where x represents the quantity demanded in units of a thousand and p is the unit price in dollars, find the equilibrium quantity and the equilibrium price 3. p-60-2x2 and p=x2 +9x + 30 Equilibrium quantity:-- Equilibrium price: 4. (1 pt) Lynbrook West, an apartment complex, has 100 two-bedroom units. The monthly profit realized from renting out x apartments is given by P(x)- 10x +1760x-50,000 dollars. How many units should be rented out to maximize the monthly rental profit? What is the maximum monthly profit realizable? Number of units: Maximum profit:- 2

Explanation / Answer

3. p = 60 - 2x2 p = x2 + 9x + 30

When the two curves intersect,

60 - 2x2 = x2 + 9x + 30

=> 3x2 + 9x - 30 = 0

=> 3(x2 + 3x - 10) = 0

=> x2 + 3x - 10 = 0

=> x2 + 5x - 2x - 10 = 0

=> x(x + 5) - 2(x + 5) = 0

=> (x - 2) (x + 5) = 0

=> x = 2 or x = -5.

Since x can't be negative, x = 2.

p = 60 - 2x2

= 60 - 2*22

= 60 - 8

= 52.

Thus the equilibrium quantity is 5 thousand units and equilibrium price is 52$.

4. P(x) = - 10x2 + 1760x - 50000

Differentiating with respect to x

=> P'(x) = - 20x + 1760

Differentiating again with respect to x

=> P''(x) = -20 < 0

Thus there is a maximum at P'(x) = 0

=> -20x + 1760 = 0

=> x = -1760 / -20

=> x = 88

Thus 88 apartments need to be rented to maximize profits.

P(88) = - 10 *882 + 1760*88 - 50000

= - 77440 + 154880 - 50000

= $27440

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