During the summer months Terry makes and sells necklaces on the beach. Terry not
ID: 2882882 • Letter: D
Question
During the summer months Terry makes and sells necklaces on the beach. Terry notices that if he lowers the price, he can sell more necklaces, and if he raises the price than he sells fewer necklaces. The table below shows how the number n of necklaces sold in one day depends on the price p (in dollars). (a) Find a linear function of the form n = c_0 + c_1p that best fits these data, using least squares. n = n(p) = 7.46-1.46p (b) Find the revenue (number of items sold times the price of each item) as a function of price p. R = R(p) = 7.46-1.46p^2 (c) If the material for each necklace costs Terry 6 dollars, find the profit (revenue minus cost of the material) as a function of price p. P = P(p) = 16.22p-1.46p^2-44.76 (d) Finally, find the price that will maximize the profit. p = 5.109589041.Explanation / Answer
(a) Here n = number of necklaces and p = price of a necklace. Mean of n values = n_mean = 24, Mean of Price = p_mean = 11.3333 Now s_xy = sum of product of (n - n_mean) and (p - p_mean) = -63. And s_xx = sum of square of (p - mean_p) = 24.6667. Using the method of least squares, slope = s_xy / s_xx = -2.5541 and the intercept = n_mean - slope * p_mean = 52.94. So using the least squares method, c_o = 59.94595 and c_1 = -2.5541 , so the equation is n = n(p) = 59.94595 -2.5541p
(b) Revenue is number of items sold times the price of each item. So R = R(p) = n(p)*p = 59.94595p -2.5541p^2
(c) If the material of each necklace costs $6, then the cost of n necklaces = 6n dollars. Now , profit = revenue - cost. Hence P = P(p) = 59.94595p -2.5541p^2 - 6n = 59.94595p -2.5541p^2 - 6(59.94595 -2.5541p) = -2.5541p^2 +75.2706p - 359.676. So, P(p) = -2.5541p^2 +75.2706p - 359.676
(d) To maximize the profit, we need to equate the derivative of P to zero and solve for p. Derivative of P(p) = 75.2706-5.1082p = 0. Solving for p = 14.73525. After rounding, this is p = $14.74. So the profit becomes maximum when the price of each necklace is $14.74
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