Could someone work this question out so I understand it. Thanks
The marginal price dp/dx at x units of demand per week is proportional to the price p. There is no weekly demand at a price of $100 per unit [p(0)=100], and there is a weekly demand of 8 units at the price of $60.83 per unit [p(8)=60.83].
A)find the price demand equation. Give an exact answer in simplified form. Round all decimal values to the nearest hundreth.
This problem can not be completed because in this case dp/dx will cancel itself out to get one...
To find the price-demand equation, we need to use the given information about the marginal price and demand. Let's denote the price as p and the demand as x.
Given that the marginal price dp/dx at x units of demand per week is proportional to the price p, we can express this relationship mathematically as:
dp/dx = k * p
where k is the proportionality constant.
Now, we are given two points of the demand and price relationship: (0, 100) and (8, 60.83). We can use these points to find the value of k.
Let's start with the first point (0, 100). Substituting the values into the equation:
dp/dx = k * 100
Now, let's consider the second point (8, 60.83):
dp/dx = k * 60.83
Since the marginal price dp/dx is the same at both points, we can equate the two expressions:
k * 100 = k * 60.83
Now, divide both sides by k:
100 = 60.83
Simplifying, we find:
k = 100/60.83 ≈ 1.642
Now that we have the value of k, we can substitute it back into one of the expressions to find the price-demand equation. Let's use the first point (0, 100):
dp/dx = 1.642 * p
Integrating both sides with respect to x:
∫ dp = ∫ 1.642 * p dx
Integrating both sides:
p = 1.642 * ∫ dx
p = 1.642x + C
To find the constant of integration, we can substitute the values of x and p from either of the given points. Let's use (0, 100):
100 = 1.642 * 0 + C
C = 100
Therefore, the price-demand equation is:
p = 1.642x + 100
Now, the question asks us to give the answer in simplified form. Since the equation is already in simplified form, our final answer is:
p = 1.642x + 100 (rounded to the nearest hundredth)