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

for a i got C(q)= (a/q) + b
C'(q)= -(a/q^2) DOLLARS PER CELL PHONE.

for b) i started off as
dC/dt =-a/q^2 dq/dt =
(-200000/q^2).100
IM LOST AND NOT SURE WHAT TO DO PLEASE HELP ME

also can you please tell me if A is right? thank u

Well, well, well! It seems like you're in a bit of a pickle here. Don't worry, my friend, the Clown Bot is here to help you out with a dash of humor! Let's break it down step by step.

a. We want to find the rate of change of C as q increases. Remember, the units of C are dollars, so we're looking for the units of the rate of change. You correctly found that C'(q) = -(a/q^2), which means the rate of change of C is in dollars per cell phone. So there you have it, the units are dollars per cell phone. Just don't go spending all those cell phone dollars in one place!

b. Now, let's tackle the second part of your question. You want to find how fast the average cost is changing when a = 200,000 and q is increasing at a rate of 100 cell phones per week. To do this, we need to find dC/dt, the rate of change of C with respect to time, when q = 5000 and dq/dt = 100.

First, plug in the given values into the formula: C = (a/q) + b. Since a = 200,000 and q = 5000, we have C = (200,000/5000) + b. Simplifying this, we get C = 40 + b.

Now, we need to find dC/dt. Since q is increasing at a rate of 100 cell phones per week (dq/dt = 100), we can use the chain rule to find dC/dt. Have no fear, the Clown Bot has the equation at hand: dC/dt = dC/dq * dq/dt.

From your calculations earlier, you found that dC/dq = -(a/q^2). Substituting the given values, we have dC/dq = -(200,000/5000^2) = -0.08 dollars per cell phone.

Now, we can multiply dC/dq by dq/dt to find dC/dt. Substituting dq/dt = 100, we get dC/dt = -0.08 * 100 = -8 dollars per week.

So, my dear friend, the average cost is decreasing at a rate of 8 dollars per week when a = 200,000 and q is increasing at a rate of 100 cell phones per week. Keep an eye on those costs and don't let them slip through your fingers!

To find the rate of change of C as q increases, you need to differentiate the equation C(q) = (a/q) + b with respect to q.

So, taking the derivative of C(q) with respect to q, we get:

C'(q) = -a/q^2

The units of C'(q) would be dollars per unit of q (cell phone in this case).

Now, let's move on to part b) of the question. We are given that a = 200000, q = 5000, and dq/dt = 100. We need to find how fast the average cost is changing, which means we need to find dC/dt.

Using the equation C(q) = (a/q) + b, we can differentiate it with respect to t (time):

dC/dt = -(a/q^2) * dq/dt

Substituting the given values, we have:

dC/dt = -(200000/5000^2) * 100

Simplifying further:

dC/dt = -(200000/25000000) * 100

dC/dt = -0.8 dollars per week

Therefore, the average cost is decreasing at a rate of 0.8 dollars per week when a = 200000, q = 5000, and production is increasing by 100 cell phones per week.

To find the rate of change of C as q increases, we need to find the derivative of C with respect to q, which is denoted as dC/dq.

a. Start with the equation for C: C = (a/q) + b
Take the derivative of both sides with respect to q:
dC/dq = d/dq((a/q) + b)
dC/dq = -a/q^2

The units for the rate of change of C as q increases can be determined by looking at the units of the variables involved. In this case, C is given in dollars, while q represents the quantity of cell phones being manufactured. Therefore, the units for dC/dq would be "dollars per cell phone."

b. To determine how fast the average cost is changing at a specific point in time, we need to substitute the given values into the expression we derived for the rate of change.

Given:
a = 200,000 (dollars)
q = 5,000 (cell phones)
dq/dt = 100 (cell phones per week)

Substitute these values into the equation:
dC/dt = (-a/q^2) * dq/dt
dC/dt = (-200,000 / (5,000)^2) * 100
dC/dt = (-200,000 / 25,000,000) * 100
dC/dt = -0.8 dollars per week

The rate of change of the average cost is -0.8 dollars per week. Since the rate is negative, it means that the average cost is decreasing.

a, No.

YOu want dC/dq (which is cost/unit)

dC/dq=-a/q^2

b) dC/dt=-a/q^2 dq/dt
dC/dt=-2E5/25E6 *1E2=-20 dollars /week