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.
B)At a demand of 25 units per week, what is the price? Round to the nearest cent. p(25) =
dp/dx = kp
points on p(X): (0,100) ; (8;60.83)
dp/dx= kp
dp/p= kdx
ln(p)=kx + c
ln(100)=k(0) + c
4.6=c
ln(60.83)=8k + 4.6
k= -0.0614
ln(p)=-.0614*25+4.6
ln p= 3.063
p= 21.39 at x=25
check my thinking.
To find the price demand equation, we can set up a proportion based on the given information.
Let's assume that the price demand equation is of the form p(x) = kx, where k is the constant of proportionality.
We are given two data points:
1. p(0) = 100, which means that when the demand is 0, the price is $100 per unit.
2. p(8) = 60.83, which means that when the demand is 8 units, the price is $60.83 per unit.
Using these two data points, we can set up the following proportion:
(60.83 - 100) / (8 - 0) = k
Simplifying the equation:
-39.17 / 8 = k
k = -4.89625
Now that we have the value of k, we can substitute it back into the price demand equation:
p(x) = -4.89625x
This is the price demand equation in simplified form.
To find the price at a demand of 25 units per week, we can substitute x = 25 into the price demand equation:
p(25) = -4.89625 * 25
p(25) = -122.40625
Rounding to the nearest cent, the price would be approximately -$122.41 per unit. Note that negative price indicates a decrease in price compared to the initial price.