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The demand function for a certain product is determined by the fact that the product of the price and the quantity demanded equals 8000. The product currently sells for ​$3.40 per unit. Suppose manufacturing costs are increasing over time at a rate of 12​% and the company plans to increase the price p at this rate as well. Find the rate of change of demand over time.

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opudodennis

Answer:

83.045( decreasing demand per unit time)

Step-by-step explanation:

let p be the price per product and q the number of units sold.

Given that p=$3.40 and q rate is 12%

-We are given that:

[tex]pq=8000\\\\p=3.4\\\\\frac{dp}{dt}=0.12[/tex]

#We differentiate using product rule to get the rate at which demand changes over time:

[tex]p\frac{dq}{dt}+q\frac{dp}{dt}=0\\\\\frac{dq}{dt}=\frac{-q\frac{dp}{dt}}{p}\\\\=-\frac{(8000/3.4)(0.12)}{3.5}\\\\=-83.045[/tex]

Hence, demand is decreasing at a rate of 83.045 units per unit time.

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