An aircraft factory manufactures airplane engines. The unit cost
C
(the cost in dollars to make each airplane engine) depends on the number of engines made. If
x
engines are made, then the unit cost is given by the function
=Cx+−0.2x248x21,748
. What is the minimum unit cost?
Do not round your answer.
C appears to be a quadratic. As with any quadratic
ax^2+bx+c
the vertex lies at x = -b/2a
So, find C(x) at that value.
To find the minimum unit cost, we need to find the value of x that minimizes the function C(x) = Cx+−0.2x248x21,748.
To do this, we can take the derivative of C(x) with respect to x and set it equal to 0 to find the critical points. Then, we can check these critical points to determine which one minimizes the function.
First, let's find the derivative of C(x):
C'(x) = C - 0.2x^2 + 48x + 21748
Next, we set C'(x) = 0 and solve for x:
C - 0.2x^2 + 48x + 21748 = 0
Now, let's solve this quadratic equation for x.
To find the minimum unit cost, we need to find the value of x that minimizes the function C(x). The function C(x) is given as:
C(x) = Cx - 0.2x^2 + 248x + 21,748
To find the minimum, we can find the derivative of the function C(x) with respect to x and set it equal to zero. Let's go step by step:
1. Take the derivative of C(x) with respect to x:
C'(x) = d/dx (Cx - 0.2x^2 + 248x + 21,748)
= C - 0.4x + 248
2. Set the derivative equal to zero and solve for x:
C'x - 0.4x + 248 = 0
Since C is not given in the question, we cannot solve for the specific value of x. However, we can find the general value of x that minimizes the unit cost.
If we assume C = 0, then the equation becomes:
-0.4x + 248 = 0
Solving for x, we have:
-0.4x = -248
x = -248 / -0.4
x = 620
Therefore, if C is assumed to be zero, the value of x that minimizes the unit cost is 620. However, keep in mind that the minimum unit cost may vary depending on the value of C.