Rate of Change of Demand

Calculus
Tutor: None Selected Time limit: 1 Day

a price (in dollars) and demand x for a product are related by

2x^2 +1xp +50p^2 = 22400

If the price is increasing at a rate of $2 per month when price is $20, find the rate of change of demand

Apr 2nd, 2015

p and x are functions of t (time), dp/dt and dx/gt their rates of change over time. 
Given that p and x are related by 2x² + xp + 50p² = 22400, 
when p = 20, x is defined by 
2x² + 20x + 20000 = 22400, (x > 0) 
x² + 10x - 1200 = 0, (x > 0) 
(x + 40)(x - 30) = 0, (x > 0) 
hence x = 30 

Differentiating each side of the equation with respect to t: 
4x(dx/dt) + p(dx/dt) + x(dp/dt) + 100p(dp/dt) = 0 
Therefore: 
dx/dt = (-x - 100p)(dp/dt)/(4x + p) 

At p = 20, x = 30 and dp/dt = 2 
Hence: 
dx/dt = (-30-2000)*2/(120 + 20)=-29

Rate of change of demand = -29 unit per month 

At the initial price of 20 dollars per unit, 
if the price increases at the rate of 2 dollars/month 
then the demand will decrease at the rate of -29 unit per month.


Apr 2nd, 2015

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