The number of retro portable CD players you are prepared to supply to a retail outlet every week is given by the formula
q = 0.1p2 + 3p
where p is the price it offers you. The retail outlet is currently offering you $100 per CD player. If the price it offers decreases at a rate of $4 per week, how will this affect the number you supply?
dq/dt = (0.2p + 3) dp/dt
so, at p=100,
dq/dt = 23 (-4) = -92
You will supply 92 fewer per week
To determine how the decrease in price offered by the retail outlet will affect the number of retro portable CD players supplied, we need to evaluate how the quantity supplied (q) changes as the price (p) decreases.
Given the formula: q = 0.1p^2 + 3p
We can substitute the current price of $100 into the equation:
q = 0.1(100)^2 + 3(100)
q = 0.1(10000) + 300
q = 1000 + 300
q = 1300
So, currently, you are supplying 1300 retro portable CD players every week to the retail outlet.
Now let's evaluate how the quantity supplied changes as the price offered decreases by $4 per week. We will calculate the quantity supplied for each subsequent decrease in price.
When the price decreases to $96 (a decrease of $4):
q = 0.1(96)^2 + 3(96)
q = 0.1(9216) + 288
q = 921.6 + 288
q = 1209.6
When the price decreases to $92 (another decrease of $4):
q = 0.1(92)^2 + 3(92)
q = 0.1(8464) + 276
q = 846.4 + 276
q = 1122.4
When the price decreases to $88 (another decrease of $4):
q = 0.1(88)^2 + 3(88)
q = 0.1(7744) + 264
q = 774.4 + 264
q = 1038.4
As you can see, as the price offered decreases by $4 per week, the quantity supplied (q) also decreases. The number of retro portable CD players supplied decreases from 1300 to approximately 1209.6, 1122.4, and 1038.4 as the price decreases to $96, $92, and $88, respectively.
To determine how the decrease in price offered by the retail outlet will affect the number of retro portable CD players you supply, we need to substitute the new price into the given formula and calculate the corresponding quantity.
Let's start by determining the new price offered by the retail outlet, considering that it decreases at a rate of $4 per week. The initial price is $100 per CD player, and it decreases by $4 per week.
If we represent the number of weeks as "w," then the new price can be calculated using the following formula:
price = initial price - rate * weeks
price = $100 - $4 * w
price = $100 - 4w
Now, let's substitute this new price into the equation q = 0.1p^2 + 3p, where p is the price. We obtain:
q = 0.1(100 - 4w)^2 + 3(100 - 4w)
Simplifying further:
q = 0.1(10000 - 800w + 16w^2) + 300 - 12w
q = 1000 - 80w + 1.6w^2 + 300 - 12w
q = 1300 - 92w + 1.6w^2
Thus, the new equation to represent the number of retro portable CD players you are prepared to supply based on the decreasing price is q = 1.6w^2 - 92w + 1300.
This equation represents the relationship between the number of CD players you are prepared to supply (q) and the number of weeks (w), considering the decreasing price offered by the retail outlet.
To determine how this decrease in price affects the quantity supplied, you can analyze the change in the equation. In particular, you can calculate the difference in the number of CD players supplied for different values of w (assuming an integer value for simplicity) and observe how it varies as w increases or decreases.