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.
dp/dx = k p
dp/p = k dx
ln p = kx + C
p = e^(kx+C) = c e^kx
p(0)= + 100
so
100 = c e^0 = c
so
p = 100 e^kx
p(8) = 60.83
60.83 = 100 e^(k*8)
ln(.6083) = 8 k
k = -.06214
so
p = 100 e^(-.06214 x)
if x = 25
p = 100 e^-(.0214*25)
p = 100 * .5857
p = $58.57
To find the price-demand equation, we can start with the given information. We know that the marginal price dp/dx at x units of demand per week is proportional to the price p.
Let's denote the constant of proportionality as k. This means that we can write the equation as:
dp/dx = k * p
To find the value of k, we can use the given information about the weekly demand and price. We know that there is no weekly demand at a price of $100 per unit (p(0)=100) and a weekly demand of 8 units at the price of $60.83 per unit (p(8)=60.83).
Let's substitute these values into the equation:
dp/dx = k * p
At x = 0, p = 100:
dp/dx = k * 100
At x = 8, p = 60.83:
dp/dx = k * 60.83
Since dp/dx is the derivative of p with respect to x, we can integrate both sides of the equation to solve for p:
∫ dp/dx dx = ∫ k * p dx
Integrating with respect to x:
∫ dp = ∫ k * p dx
Applying the definite integral between the values of x = 0 to x = 8:
∫dp |[0, 8] = ∫k * p dx |[0, 8]
This gives us:
p(8) - p(0) = k * ∫[0,8] p dx
Substituting the given values:
60.83 - 100 = k * ∫[0,8] p dx
-39.17 = k * ∫[0,8] p dx (Equation 1)
To find the price-demand equation, we need to solve this equation. However, we need more information or an expression for the definite integral to proceed further.
Moving on to part B:
To find the price at a demand of 25 units per week, we can use the price-demand equation once we have obtained it in part A.
To find the price demand equation, we can use the given information. The problem states that the marginal price dp/dx at x units of demand per week is proportional to the price p. This can be written as:
dp/dx = kp
where k is the proportionality constant.
We are also given two data points: p(0) = 100 and p(8) = 60.83.
To find the value of k, we can substitute the first data point (p = 100, x = 0) into the equation:
dp/dx = kp
100k = k(0)
100k = 0
Since k cannot be zero (otherwise there will be no proportional relationship), we conclude that k is not equal to zero.
Now, let's substitute the second data point (p = 60.83, x = 8) into the equation:
dp/dx = kp
60.83k = k(8)
60.83k = 8k
Dividing both sides of the equation by k (remember k is not equal to zero):
60.83 = 8
Now we know that 60.83 = 8. Thus, we have determined the value of k, which is 8.
Now we can rewrite the equation dp/dx = kp as:
dp/dx = 8p
To solve this first-order linear differential equation, we can use separation of variables:
dp/p = 8dx
Integrating both sides:
∫(1/p) dp = ∫8 dx
ln|p| = 8x + C1 (where C1 is the constant of integration)
Now, we can exponentiate both sides:
|p| = e^(8x + C1)
Since p represents price, it must be positive. Therefore, we can remove the absolute value:
p = e^(8x + C1)
Finally, let's use the initial condition p(0) = 100 to find the value of the constant C1:
100 = e^(8(0) + C1)
100 = e^C1
Taking the natural logarithm of both sides:
ln(100) = C1
C1 ≈ 4.60517
Now we have the price demand equation:
p = e^(8x + 4.60517)
This equation gives the price p as a function of the demand x.
Now let's move on to part B of the question.
To find the price at a demand of 25 units per week, we can substitute x = 25 into the price demand equation:
p = e^(8x + 4.60517)
p = e^(8(25) + 4.60517)
Using a scientific calculator or software, we can calculate the expression inside the exponential function:
p ≈ e^206.20517
Rounding the result to the nearest cent, we have the final answer:
p ≈ $8874236546572.31 (rounded to the nearest cent)