a distributor of apple juice has 5000 bottle in the store that it wishes to distribute in a month. from experience it is known that demand D (in number of bottles) is given by D = -2000p² + 2000p + 1700. the price per bottle that will result zero inventory is
let D=5000
D=-2000p^2+2000p+17000
5000=-2000p^2+2000p+17000
divided by 1000
5=-2p^2+2p+17
5-17=-2p^2+2p
-12=-2p^2+2p
2p^2-2p-12=0
divided by 2
p^2-p-6=0
p^2-3p+2p-6=0
p(p-3)+2(p-3)=0
(p-3) (p+2)=0
p-3=0 p+2=0
p=3 p=-2
To find the price per bottle that will result in zero inventory, we need to determine the price (p) when the demand (D) is equal to the current stock of 5000 bottles.
Given that the demand equation is D = -2000p² + 2000p + 1700, we can set D equal to 5000 to find the price that will result in zero inventory.
5000 = -2000p² + 2000p + 1700
To solve this equation, we can rearrange it to form a quadratic equation:
-2000p² + 2000p + 1700 - 5000 = 0
Simplifying the equation further:
-2000p² + 2000p - 3300 = 0
To solve this quadratic equation, we can apply the quadratic formula:
p = (-b ± √(b² - 4ac)) / 2a
Here, a = -2000, b = 2000, and c = -3300.
Plugging these values into the quadratic formula:
p = (-2000 ± √(2000² - 4*(-2000)*(-3300))) / (2*(-2000))
Simplifying the equation further:
p = (-2000 ± √(4000000 - (4*2000*3300))) / (-4000)
p = (-2000 ± √(4000000 + 26400000)) / (-4000)
p = (-2000 ± √(30400000)) / (-4000)
p = (-2000 ± 5524.37) / (-4000)
Now we have two possible solutions for p:
p₁ = (-2000 + 5524.37) / (-4000)
p₁ = 3524.37 / -4000
p₁ ≈ -0.881
p₂ = (-2000 - 5524.37) / (-4000)
p₂ = -7524.37 / -4000
p₂ ≈ 1.881
Since we can't have a negative price, the price per bottle resulting in zero inventory would be approximately $1.881.
To find the price per bottle that will result in zero inventory, we need to find the demand (D) when the inventory is completely distributed.
The inventory is represented by the number of bottles in the store, which is initially 5000. As the distributor sells the bottles, the inventory decreases, and at the end of the month, the inventory should reach zero.
We also know that the demand (D) is given by the equation D = -2000p² + 2000p + 1700, where "p" represents the price per bottle.
To find the price per bottle that will result in zero inventory, we need to solve the equation D = 0.
-2000p² + 2000p + 1700 = 0
This is a quadratic equation. We can solve it by factoring or by using the quadratic formula.
Since the equation is not easily factorable, we will use the quadratic formula:
p = (-b ± √(b² - 4ac)) / (2a)
In our equation, a = -2000, b = 2000, and c = 1700.
Using the quadratic formula, we get:
p = (-2000 ± √(2000² - 4(-2000)(1700))) / (2(-2000))
p = (-2000 ± √(4000000 + 13600000)) / (-4000)
p = (-2000 ± √(17600000)) / (-4000)
p = (-2000 ± 4200) / (-4000)
Simplifying further, we have:
p = (-2000 + 4200) / (-4000) or p = (-2000 - 4200) / (-4000)
p = 2200 / (-4000) or p = -6200 / (-4000)
p = -0.55 or p = 1.55
So, the price per bottle that will result in zero inventory is either $0.55 or $1.55.
just solve for p in
-2000p² + 2000p + 1700 = 5000
The problem is that D has a maximum of 2200 when p = 0.5
I suspect a typo somewhere. Is D the monthly demand, or the weekly demand?