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)