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Math calculus derrivative 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.

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Mark M. | Math Tutor--High School/College levelsMath Tutor--High School/College levels
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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.  
 

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Michael J. | Effective High School STEM Tutor & CUNY Math Peer LeaderEffective High School STEM Tutor & CUNY ...
5.0 5.0 (5 lesson ratings) (5)
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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.