Cost: C^2=x^2+98*sq rt of x + 57
Revenue:890(x-4)^2+29R^2=26,100
Find the marginal cost dC/dx at x=4
Assuming you mean
c^2 = x^2 + 98√(x+57)
then
c^2(4) = 4^2 + 98√61
= 16+98√61
c(4) = 27.95
2c dc/dx = 2x + 49/√(x+57)
at x=4, we have
55.90 dc/dx = 8+49/√61
dc/dx = 0.255
Of course, I might have misread the function, with all those pesky words instead of nice math notation ...
To find the marginal cost, we need to differentiate the cost function with respect to "x" and evaluate it at x = 4.
Given:
Cost function: C^2 = x^2 + 98 * √x + 57
Step 1: Differentiate the cost function with respect to "x".
To differentiate C^2, we need to use the chain rule:
d(C^2)/dx = 2C * dC/dx
Differentiating x^2 gives us 2x.
Differentiating 98 * √x gives us (98/2) * (x^(-1/2)) = 49/x^(1/2)
Differentiating 57 gives us 0 (as it is a constant).
So, the differentiation of the cost function d(C^2)/dx is:
2C * dC/dx = 2x + 49/x^(1/2)
Step 2: Substitute x = 4 into the derivative expression.
Now, substitute x = 4 in the expression we derived above:
2C * dC/dx = 2(4) + 49/(4)^(1/2)
⇒ 2C * dC/dx = 8 + 49/2
⇒ 2C * dC/dx = 8 + 24.5
⇒ 2C * dC/dx = 32.5
Step 3: Solve for dC/dx.
Divide both sides of the equation by 2C:
dC/dx = (32.5)/(2C)
To find dC/dx at x = 4, we need to determine the value of C. However, the value of C is not given in the problem statement. You will need to provide the value of C to find the exact value of marginal cost dC/dx at x = 4.