Question

Price-Supply Equation The number of bicycle helmets a retail chain is willing to sell per week at a price of $p is given by x = ap+b- where a = 78, b = 24, and c = 384. Find the instantaneous rate of change of the supply with respect to price when the price is $74. Round to the nearest hundredth (2 decimal places). helmets per dollar

Answer 1

The quantity demanded and supplied at a price of $210 is 16404 helmets. At a price of approximately $613.44, the quantity supplied is equal to 48,000. The equilibrium price and quantity can be determined by finding the intersection of the demand and supply curves.

Step-by-step explanation:

a. To find the quantity demanded and supplied at a price of $210, substitute p = 210 into the given equation. From the equation x = ap+b, we have:

x = 78(210)+24

x = 16380+24

x = 16404

So, the quantity demanded and supplied at a price of $210 is 16404 helmets.

b. To find the price at which the quantity supplied is equal to 48,000, we need to solve the equation x = ap+b for p. Substitute x = 48,000 into the equation and solve for p:

48000 = 78p+24

47876 = 78p

p = 47876/78

p ≈ 613.44

So, the price at which the quantity supplied is equal to 48,000 is approximately $613.44.

c. The demand curve and supply curve can be graphed on a coordinate plane, with the quantity on the x-axis and the price on the y-axis. The equilibrium price and quantity can be determined by finding the point at which the demand curve and supply curve intersect. From the table, the equilibrium price and quantity are the values at which the quantity demanded and supplied are equal.

d. To find the quantities demanded and supplied at a price of $120, substitute p = 120 into the given equation. From the equation x = ap+b, we have:

x = 78(120)+24

x = 9360+24

x = 9384

So, the quantities demanded and supplied at a price of $120 is 9384 helmets. Since this is less than the equilibrium quantity, a shortage would exist. The shortage would be the difference between the quantity demanded and supplied at a price of $120 and the equilibrium quantity.