The research department for a hardware chain has determined that at one store the marginal price of x boxes per week of a particular type of nail is:
p'(x)=-4000/(2x+15)^3
Find the demand equation if the weekly demand for this type of nails is 10 boxes when the price of a box of nails is $4.
Good
To find the demand equation, we need to integrate the marginal price function, p'(x), to get the price function, p(x).
Given: p'(x) = -4000/(2x+15)^3
Step 1: Integrate p'(x) to find p(x).
To integrate the function p'(x), we need to use the power rule for integration. The power rule states that the integral of x^n is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.
In this case, we have p'(x) = -4000/(2x+15)^3, so n = -3. Applying the power rule, we get:
∫ p'(x) dx = ∫ (-4000/(2x+15)^3) dx
= -4000 * ∫ (1/(2x+15)^3) dx
Step 2: Simplify and solve the integral
To integrate 1/(2x+15)^3, we can make a substitution u = 2x + 15, du = 2dx. By doing this substitution, we can simplify the integral.
∫ (1/(2x+15)^3) dx = (1/2) * ∫ (1/u^3) du
= (1/2) * (-1/2u^2) + C
= -1/(4u^2) + C
= -1/[4(2x+15)^2] + C
Step 3: Find the constant of integration
To find the constant of integration (C), we can use the given information that the weekly demand for this type of nails is 10 boxes when the price of a box of nails is $4.
Let's substitute x = 10 and p(x) = 4 into the derived equation and solve for C.
p(x) = -1/[4(2x+15)^2] + C
Substituting values:
4 = -1/[4(2 * 10 + 15)^2] + C
4 = -1/(4 * 35^2) + C
4 = -1/4900 + C
C = 4 + 1/4900
Therefore, the constant of integration (C) is 4 + 1/4900.
Step 4: Write the demand equation
Now that we know the value of the constant of integration (C), we can write the demand equation.
p(x) = -1/[4(2x+15)^2] + (4 + 1/4900)
So, the demand equation for this type of nails is p(x) = -1/[4(2x+15)^2] + (4 + 1/4900).