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. To find the rate of change of C as q increases, we need to differentiate the equation C = (a/q) + b with respect to q.
Using the power rule, the derivative of (a/q) with respect to q is -a/q^2. The derivative of b with respect to q is 0 since b is a constant.
Therefore, the derivative of C with respect to q is -a/q^2.
The units of this derivative are in dollars per unit of q squared (dollars/q^2).
b. We are given that a = 200,000, q = 5000, and dq/dt = 100.
To find how fast the average cost is changing, we substitute these values into the derivative we found in part a:
Rate of change = (-a/q^2) * dq/dt
= (-200,000/(5000)^2) * 100
= -0.8 dollars per week
Since the rate of change is negative (-0.8), the average cost is decreasing.
a. To find the rate of change of C as q increases, we need to find the derivative of the function C with respect to q. The equation is given by C = (a/q) + b.
Let's differentiate both sides of the equation with respect to q:
dC/dq = d/dq (a/q) + d/dq (b)
To differentiate (a/q) with respect to q, we can use the power rule:
d/dq (a/q) = (-a/q^2)
Since b is a constant, its derivative with respect to q is zero:
d/dq (b) = 0
Now, let's substitute these derivatives back into the equation:
dC/dq = (-a/q^2) + 0
Simplifying further, we get:
dC/dq = -a/q^2
The units of C are given in dollars, and q is the quantity of cell phones. Therefore, the units of the rate of change of C with respect to q will be dollars per (cell phone)^2.
b. To find how fast the average cost is changing, we need to calculate the derivative of C with respect to time (t), and then evaluate it at the given values.
Given: a = 200,000, q = 5000, and dq/dt = 100.
Using the equation C = (a/q) + b, let's differentiate it with respect to t:
dC/dt = d/dt (a/q) + d/dt (b)
To differentiate (a/q) with respect to t, we'll use the quotient rule:
d/dt (a/q) = (a * d/dt (q) - q * d/dt (a)) / (q^2)
Since a is a constant, its derivative with respect to t is zero:
d/dt (a) = 0
Using the given rate at which q is changing, dq/dt = 100:
d/dt (q) = 100
Now, let's substitute these values back into the equation:
dC/dt = (0 * 100 - 5000 * 0) / (5000^2)
Simplifying further, we get:
dC/dt = 0 / 25,000,000
Therefore, the rate at which the average cost is changing is zero dollars per week. This means that the average cost is not changing with time.