# A company selling widgets has found that the number of items sold, x, depends upon the price, p at which they're sold, according to the equation x=80000/(0.4p+1)^2

Due to inflation and increasing health benefit costs, the company has been increasing the price by \$0.03 per month. Find the rate at which revenue is changing when the company is selling widgets at \$6 each.

Revenue is decreasing by how much dollars per month??

## I did this for you yesterday. Did you even bother to check?

revenue = price * quantity, so the revenue r(p) is
r = p*80000/(0.4p+1)^2 = 2,000,000p/(2p+5)^2
dr/dt = -2,000,000 (2p-5)/(2p+5)^3

## Ok actually i got it. Thank you so much! i just had to multiple my previous answer with the dp/dt.

(-2849.58274)(0.03)

So it is decreasing by \$85.4874822 dollars per month.

## What were the steps to get the final equation?

r = p*80000/(0.4p+1)^2
(did you have to multiply the top and bottom of the fraction by 25)
r = 2,000,000p/(2p+5)^2
dr/dt = -2,000,000 (2p-5)/(2p+5)^3

## To find the rate at which revenue is changing, we need to calculate the derivative of the revenue function with respect to time. The revenue, R, can be calculated by multiplying the number of items sold, x, by the price, p.

Given the equation for the number of items sold: x = 80000/(0.4p + 1)^2

We can substitute this into the revenue equation: R = xp

Substituting x into R, we get: R = (80000/(0.4p + 1)^2) * p

Now, we need to find the rate at which revenue is changing with respect to time, t. Since we are given that the price is increasing by \$0.03 per month, we can let p = 6 + 0.03t.

Now, substitute p = 6 + 0.03t into R: R = (80000/(0.4(6 + 0.03t) + 1)^2) * (6 + 0.03t)

Now that we have the revenue equation, R, in terms of t (time), we can differentiate it with respect to t to find the rate at which revenue is changing.

Taking the derivative, we get: dR/dt = (80000/(0.4(6 + 0.03t) + 1)^2) * (0.03)

Simplifying further, we obtain: dR/dt = (2400000/(0.4(6 + 0.03t) + 1)^2)

Now, we can substitute the given price, p = 6, into the equation to find the rate at which revenue is changing when the company is selling widgets at \$6 each.

Substituting p = 6, we get: dR/dt = (2400000/(0.4(6 + 0.03t) + 1)^2)

Now, we can calculate the rate at which revenue is changing by evaluating this expression for the given values of p and t.

## i did check, but it wrong. Maybe I'm plugging in the wrong numbers? I'm also confused on how you got the revenue equation. Would it be possible for you to show me step by step? i really do appreciate the help!

Here is what I did based on what you told me.
-2,000,000(2(6)-5)/(2(6)+5)^3
which equals to -2849.58274

What am I doing wrong? is it the numbers I'm plugging in? or the calculation? I'm confused.