[CALCULUS RELATED RATES] A supplier produces n thousand boxes of orange daily when it is sold at p dollars per box. Suppose the relationship between n and p is (continuation in description)

Question

A supplier produces n thousand boxes of orange daily when it is sold at p dollars per box. Suppose the relationship between n and p is;

pn-3p-100n-100=0

If the daily supply is decreasing by 250 boxes per day, at what rate is the price changing when the daily supply is 5000 boxes?

Answer 1

If pn-3p-100n-100=0 then when n = 5 (5 thousand boxes), we can determine the price at that quantity:

pn - 3p - 100n - 100 = 0

pn - 3p = 100n + 100

p(n - 3) = 100n + 100

p = (100n + 100)/n - 3)

p = (100•5 + 100)/(5 - 3)

p = (500 + 100)/2

p = 300 ($300 per thousand boxes)

You want to take the derivative with respect to time ("per day" means a rate with respect to time).

For the first term (pn) you will need to use the product rule because it is the product of p and n. The product rule for this would be (pn)' = p'n + pn' where p' = dp/dt and n' = dn/dt

For the second term (-3p) you must use the chain rule because "p" is changing with respect to time so (-3p)' = -3•p' where p' is again dp/dt.

For the third term (-100n), it's just as above so (-100n)' = -100•n' where n' = dn/dt

The derivative of the constants (100 and 0) are zero.

So (pn-3p-100n-100=0)' = p'n + pn' - 3•p' - 100•n' = 0

You are solving for p' (or pd/dt) so:

p'n + pn' - 3•p' - 100•n' = 0

p'n - 3•p' = 100•n' - pn'

p'(n - 3) = 100•n' - pn'

p' = (100•n' - pn')/(n - 3)

We are told that n' (or dn/dt) = -0.250 (250 = 0.25 thousand and negative because it's going down).

We are told that n = 5 (5 thousand)

And we solved for p such that p = 300

So plugging those in we get:

p' = dp/dt = (100•(-0.250) - (300)(-0.250))/(5 - 3)

p' = dp/dt = (-25 + 75)/2

p' = dp/dt = 50/2 = 25 (increasing at a rate of $25 per day)

Answer 2

Find the price when supply = 5,000: 5p - 3p - 500 - 100 = 0. ; 2p = 600. ; p = 300. Given: dn/dt = -.25

Differentiate implicitly with respect to time:

p·dn/dt + n·dp/dt - 3dp/dt - 100dn/dt = 0

300(-.25) + 5 dp/dt - 3 dp/dt + 25 = 0

2 dp/dt = 50

dp/dt = 25.

The price / box is increasing at a rate of $25 / box.