Math calculus derrivative question

Question

The number x of printer cartridges that a store will sell per week and their price p (in dollars) are related by the equation
x^2 = 1680 − 5p^2
If the price is falling at the rate of $1 per week, find how the sales will change if the current price is $16.

Answer 1

x² = 1680 - 5p²

 

2x(dx/dt) = -10p(dp/dt)

 

Since dp/dt = -1, we have 2x(dx/dt) = 10p

 

When p = 16, x² = 1680 - 5(16)²

 

                     x² = 1680 - 1280

 

                     x² = 400   x = 20

 

So, 40(dx/dt) = 10(16)

 

           dx/dt = 160/40 = 4 printer cartridges per week

 

When p = $16, the number of cartridges sold is increasing at the rate of 4 cartridges per week.  

 

Answer 2

Take the derivative of both sides of the equation.

 

2x = -10pp'

 

 

Solve for p'.

 

-x / (5p) = p'

 

 

Set p' equal to -1 because the rate is falling.

 

-x / (5p) = -1

 

 

Solve for x when p=16.  That is all you have to do.