A manufacturer produces bolts of fabric with a fixed width. The quantity q of this fabric (measured in yards) that is sold is a function of the selling price p (in dollars per yard), so we can write q = f(p).Then the total revenue earned with selling price p is R(p) = pf(p).Given f(16) = 8000and f '(16) = −330,find R'(16).

help?

Well, this is quite a mathematical question! Let's break it down.

First, we know that q = f(p) represents the quantity of fabric sold as a function of the selling price p.

We are also given that f(16) = 8000, meaning that when the selling price is $16 per yard, 8000 yards of fabric are sold.

Next, we are given f'(16) = -330, which tells us that at a selling price of $16 per yard, the rate of change of the quantity sold with respect to the selling price is -330 yards per dollar.

Now, we need to find R'(16), which represents the rate of change of the total revenue with respect to the selling price at p = 16.

The total revenue is given by R(p) = pf(p), where p is the selling price and f(p) is the quantity of fabric sold. We can differentiate this equation with respect to p to find R'(p).

Using the product rule, we have:

R'(p) = f(p) + pf'(p)

Let's substitute p = 16 into this equation:

R'(16) = f(16) + 16f'(16)

Substituting the given values:

R'(16) = 8000 + 16(-330)

Now, we can compute the value of R'(16):

R'(16) = 8000 - 5280

R'(16) = 2720

So, the rate of change of the total revenue at a selling price of $16 per yard is 2720 dollars per yard.

Hope that helps! Remember, if you need any more assistance, I'll be here, juggling answers for you!

To find R'(16), we need to differentiate the function R(p) with respect to p and then evaluate it at p = 16.

First, let's differentiate R(p) = pf(p) using the product rule of differentiation:

R'(p) = p * f'(p) + f(p)

Now, we'll substitute p = 16 into this equation and use the given information:

R'(16) = 16 * f'(16) + f(16)

Given f(16) = 8000 and f'(16) = -330:

R'(16) = 16 * (-330) + 8000

Simplifying this expression gives us the value of R'(16):

R'(16) = -5280 + 8000

R'(16) = 2720

Therefore, R'(16) = 2720.

To find R'(16), we need to differentiate the total revenue function R(p) with respect to p and then evaluate it at p = 16.

First, let's differentiate R(p) = pf(p) using the product rule.

R'(p) = p * f'(p) + f(p) * 1

Now, we are given f(16) = 8000 and f'(16) = -330.

Substituting these values into R'(p), we get:

R'(16) = 16 * (-330) + 8000 * 1

R'(16) = -5280 + 8000

R'(16) = 2720

Therefore, R'(16) is equal to 2720.

R(p)=p(f(p))

R'= p' * f(p) + p f'(p)

But dp/dp=p'=1
so you have a simple equation to solve for R'(16)