Could someone answer this question 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 neares hundreth.
B)At a demand of 25 units per week, what is the price?Round to the nearest cent.
A) To find the price demand equation, we need to find the constant of proportionality.
Step 1: Set up the proportion using the given information:
dp/dx = kp
Step 2: Solve for k by plugging in the given data points:
When x=0, p=100:
dp/dx = k(100)
k = dp/dx / 100
When x=8, p=60.83:
dp/dx = k(60.83)
k = dp/dx / 60.83
Step 3: Set the two expressions for k equal to each other and solve for dp/dx:
dp/dx / 100 = dp/dx / 60.83
Step 4: Solve for dp/dx:
60.83(dp/dx) = 100(dp/dx)
60.83 = 100(dp/dx)
dp/dx = 60.83 / 100
dp/dx = 0.6083
Step 5: Integrate to find the price demand equation:
∫ dp = ∫ 0.6083 dx
p = 0.6083x + C
Step 6: Use the given data point (x=0, p=100) to find the constant C:
100 = 0.6083(0) + C
C = 100
Step 7: Substitute the value of C back into the equation:
p = 0.6083x + 100
Therefore, the price demand equation is p = 0.6083x + 100.
B) To find the price at a demand of 25 units per week, substitute x=25 into the price demand equation:
p = 0.6083(25) + 100
p = 15.2075 + 100
p ≈ 115.21
Therefore, at a demand of 25 units per week, the price is approximately $115.21.