##### Rate of Change of Demand

label Calculus
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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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Apr 2nd, 2015
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Apr 2nd, 2015
Oct 21st, 2017
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