How do you solve this pesky related rates problem? (Daily Supply)

Question

Hi. I am stuck trying to solve this problem:

In a certain market, x thousand crates of apples are supplied daily. Each crate costs p dollars, and the supply equation is

px - 20p - 3x + 105 = 0.

If the daily supply decreases at the rate of 250 crates per day, at what rate will the price change when the daily supply crates reaches 5000 crates?

It would be appreciated if you also explained what you are doing (calculus noob here)

Thanks in advance!

Answer 1

yeah we all got burned on it...

the independant variable x , for the # of crates in measure in THOUSANDS..

the daily supply decreases at the rate of 250 creates per day,

so dx/dt = -0.25 = -250/1000

LIkewise, the supply goal is x=5, or 5000 crates.

differentiating...

p dx/dt + dp/dt x - 20 dp/dt - 3 dx/dt = 0

Note that when x= 5,

5p - 20*p - 3(5) + 105 = 0

-15p - 15 + 105 = 0

-15p + 90 = 0

p = -90/-15 = 6

so the price is $6 when there are 5000 crates in inventory,

that is when x= 5..

Plugging in..

6 * -0.25 + dp/dt * 5 - 20 * dp/dt - 3 * -0.25 = 0

-1.5 + dp/dt * 5 - 20 * dp/dt + 0.75 = 0

-15 * dp/dt - 0.75 = 0

dp/dt =- 0.75/15 = -0.05

so the price is dropping by a nickel