The demand equation for a certain brand of GPS Navigator is x + 3p - 565 = 0, where x is the quantity demanded per week and p is the wholesale unit price in dollars.
The supply equation is x - 16p + 480 = 0, where x is the quantity the supplier will make available in the market when the wholesale price is p dollars each. Find the equilibrium quantity and the equilibrium price for the GPS Navigators.
a. equilibrium quantity 2,000 units; equilibrium price $55
b. equilibrium quantity 2,000 units; equilibrium price $40
c. equilibrium quantity 400 units; equilibrium price $55
d. equilibrium quantity 400 units; equilibrium price $40
Well, it seems like we've got a good old-fashioned supply and demand problem here! Let's see if we can clown our way to the right answer!
To find the equilibrium quantity and price, we need to set the demand equation equal to the supply equation. So, let's do some math magic:
x + 3p - 565 = x - 16p + 480
Now, let's simplify this equation:
19p = 1,045
Dividing both sides by 19, we find:
p = 55
Now that we've found the equilibrium price, we can substitute it back into either the demand or supply equation to find the equilibrium quantity. Let's use the demand equation:
x + 3(55) - 565 = 0
Simplifying this equation, we find:
x = 2,000
So, the correct answer is option a. The equilibrium quantity is 2,000 units, and the equilibrium price is $55.
And just like that, we've navigated our way to the right answer! Are you impressed? I'm GPS-tounded myself!
To find the equilibrium quantity and price for the GPS Navigators, we need to set the demand equation equal to the supply equation and solve for x (quantity) and p (price).
First, set the demand equation equal to the supply equation:
x + 3p - 565 = x - 16p + 480
Simplify the equation:
19p = 1,045
Divide both sides of the equation by 19:
p = 1,045 / 19
Calculate the price:
p ≈ 55
Substitute the calculated price back into either the demand or supply equation to find the equilibrium quantity:
x + 3(55) - 565 = 0
Simplify the equation:
x + 165 - 565 = 0
Combine like terms:
x - 400 = 0
Add 400 to both sides of the equation:
x = 400
Therefore, the equilibrium quantity is 400 units and the equilibrium price is $55.
Therefore, the correct answer is option c: equilibrium quantity 400 units; equilibrium price $55.
To find the equilibrium quantity and price for the GPS Navigators, we need to solve the system of equations formed by the demand equation and the supply equation.
First, let's write down the demand and supply equations again:
Demand equation: x + 3p - 565 = 0
Supply equation: x - 16p + 480 = 0
To find the equilibrium quantity and price, we need to find the values of x (quantity) and p (price) that satisfy both equations simultaneously.
To solve this system of equations, we can use the method of substitution or elimination.
Let's use the substitution method:
1. Solve one equation for one variable in terms of the other variable.
We'll solve the demand equation for x in terms of p:
x = 565 - 3p
2. Substitute this expression for x in the supply equation:
565 - 3p - 16p + 480 = 0
3. Simplify and solve for p:
565 + 480 = 3p + 16p
1045 = 19p
p = 1045/19
p ≈ 55
Now that we have the value of p, let's substitute it back into one of the original equations to find the value of x:
Using the demand equation:
x + 3(55) - 565 = 0
x + 165 - 565 = 0
x = 565 - 165
x = 400
Therefore, at equilibrium, the quantity demanded (x) is 400 units, and the wholesale unit price (p) is approximately $55.
So the correct answer is option c. equilibrium quantity 400 units; equilibrium price $55.