The average cost per item, C, in dollars, of manufacturing a quantity q of cell phones is given by
C= (a/q) + b
where a and b are positive constants
a.Find the rate of change of C as q increases. What are its units?
b. Currently, 5000 phones are being made and production is increasing at a rate of 100 cell phones per week. How fast is the average cost changing if a = 200000? Is the average cost increasing or decreasing?
a. C'=-a/q^2 Units: dollars per cell phone unit
a. To find the rate of change of C as q increases, we need to differentiate the function C with respect to q. In this case, since C is a function of q, we can use the quotient rule of differentiation.
First, let's differentiate the function C = (a/q) + b. Applying the quotient rule, we have:
dC/dq = [(q*d(a/q) - (a/q)*dq)/q^2]
To simplify this expression, we can expand and cancel terms:
dC/dq = [d(a/q) - (a/q^2)*dq]/q
= [(a*dq - a*q*dq)/q^2]/q
= [a*dq - a*q*dq]/q^3
= a*(1/q - q/q^3)
= a/q^2 - a/q^4
Thus, the rate of change of C as q increases is given by dC/dq = a/q^2 - a/q^4.
The units of the rate of change of C can be determined by considering the units of a and q in the given equation. Since a is in dollars and q represents the quantity of cell phones, the rate of change of C will be in dollars per unit of q. Therefore, the units of the rate of change of C are dollars per cell phone.
b. In this case, we are given that a = 200000, q = 5000, and dq/dt = 100 (where dq/dt represents the rate of change of q, in this case, 100 cell phones per week).
To find how fast the average cost is changing, we need to substitute the given values into the equation for the rate of change of C that we derived earlier:
dC/dt = a/q^2 - a/q^4
= (200000)/(5000)^2 - (200000)/(5000)^4
= 0.08 - 0.000008
= 0.079992
So, the average cost is changing at a rate of approximately 0.079992 dollars per cell phone per week.
To determine whether the average cost is increasing or decreasing, we need to consider the sign of the rate of change. In this case, the rate of change is positive (0.079992 dollars per cell phone per week), which means the average cost is increasing.