Posted: 2/27/2008 12:57:24 PM EDT
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How do I solve this problem, and also whats the answer? When the price of a certain commodity is p dollars per unit, customers demand x hundred units of the commodity, where x2 + 6p - p3=13 How fast is the demand x changing with respect to time when the price is $7 per unit and is changing at a rate of 0.23 dollars per month? Enter your answer to two decimal places. |
The equation is a parabola. Take its Derivative with respect to Time. Since the Price is Fluctuating it is not constant. 2xX' + 6p' - 3p^2p' = 0 (Chain Rule) Solving for x: 2xX' = 3p^2p' - 6p' --> X' = (3p^2p' - 6p')/2x pluging in the values p' = 0.23 dollars per month p = 7 dollars x = xhundred we get: X' = (3*7*7*0.23 - 6*0.23)/2xhundred --> X' = (33.81 - 1.38)/2xhundred -->X' = 0.16215/x where x is some number (I already divided the numerator by 100). Now we solve for X: x2 + 6p - p3=13 since p = 7 dollars So -->x^2 + 6*7 - 7^3 = 13 -->x^2 + 42 - 343 = 13 --> x^2 = 314 --> x = 17.72 We now plug this value for x into the equation: X' = 0.16215/x -->X' = 0.16215/17.72 -->X' = 0.00915 in Dollars per Month Note: X' = dx/dt and p' = dp/dt |