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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